View Full Version : Russell's Paradox and the Excluded-Middle reasoning
By tautology x = x means: x is itself, otherwise we cannot talk about x.
Now we can ask if a teotology is also recursive, for example: x = x = x = ...
If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.
So, Russel's paradox is like if by teotology we ask if x is not_x or x = not_x , which is a meaningless question trough an exluded-middle point of view.
Because our one_step_recursion cannot be done, we have no product of tautology and we can conclude that the whole idea of Russel's paradox simply does not hold in an excluded-middle logical reasoning.
By the way, in Russell's paradox x is not the set, but the word "contain".
So I hope that you agree with me that the question: Is "contain" = "do_not_contain", is meaningless through an excluded-middle reasoning.
The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.
The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
(remark: the existence of a set is not dependent on its content for example:
The empty set exists as a framework which examines the abstract idea of "emptiness".
In short, only the name of the set depends on its property, but not its own existence as a framework.)
Therefore there is nothing here that can be found as a state of a paradox.
What do you think?
ram2048
Jun12-04, 08:13 AM
would help if you laid out the paradox instead of just talking about it and assuming people know what the heck you're talking about :D
http://plato.stanford.edu/entries/russell-paradox/
Well, first off, the adverb "not" is only applicable to logical propositions/statements/formulae. Saying "not x" is meaningless unless x is a proposition.
Secondly, "x = not x" is most certainly a meaningful statement. (If, of course, x is a proposition) It's just false.
Well, first off, the adverb "not" is only applicable to logical propositions/statements/formulae. Saying "not x" is meaningless unless x is a proposition.
Then please show us how we use the proper formal way to notate x and not_x where x is a general notation for any thing, which is not a logical condition, Thank you.
Secondly, "x = not x" is not a fact in Russell's Paradox, but a sort of a question which its result has logically be checked and determinate by us.
So, first we have to check if this paradox can really exists before we starting our x not_x circular situation, which leads us to conclude that we are in an impossible excluded-middle state.
So, please read again my first post and try to understand its tautology/recursion idea, before you air your view about it.
Well, first off, the adverb "not" is only applicable to logical propositions/statements/formulae. Saying "not x" is meaningless unless x is a proposition.
Maybe irrelevant, but there's also a "not" operator in computing which works on numbers represented by bits (one's complement operator).
edit: oops, I just checked with the windows calculator, it turns out it just makes a number negative
Well, the first thing is to figure out just what you mean by "not x". As I was composing on my reply, I hit upon something grammatical that may have led you to make statements such as you have been doing.
If we take the statement "y is z", we can break it up into three parts; we have two objects, "y" and "z", and we have a relation, "is".
The negation of this statement is "y is not z". Grammatically, the right way to parse this phrase is that "not" modifies "is". In other words when we break this statement into three parts, we get two objects, "y" and "z", and we have a relation, "is not".
It would be incorrect to interpret the phrase "y is not z" as having two objects, "y" and "not z", being connected by "is".
That is why, symbolically, we write the phrase "y is not z" as something like y != z or y \neq z; the relation means "is not".
(maybe things would be clearer if, instead of "is", you use "is equal to"? I think the latter is somewhat more proper)
Hurkyl,
x is not_x in an excluded-middle reasoning system, is the reason why we are calling it a paradox.
I say that the situation x is not_x in this case simply does not exist, therefore it is avoided before we can conclude that x is not_x is a paradox in an excluded-middle system.
So, please read again post #1, thank you.
Sorry for interrupting again.
x = not x looks like x = -x
by substituting we get:
x = - ( - ( - (.....- x)
which is like:
+1 * -1 * +1 * -1 * ...... (* denotes multiplication)
Looks like a paradox.
Do you think this is relevant?
master_coda
Jun12-04, 07:53 PM
Sorry for interrupting again.
x = not x looks like x = -x
by substituting we get:
x = - ( - ( - (.....- x)
which is like:
+1 * -1 * +1 * -1 * ...... (* denotes multiplication)
Looks like a paradox.
Do you think this is relevant?
How is x = -x a paradox? 0 = -0, after all.
How is x = -x a paradox? 0 = -0, after all.
hmm. do all mathematicians agree -0 = +0? (possibly a stupid question)
But back to the topic: maybe in this context it means x doesn't exist.
master_coda
Jun12-04, 08:31 PM
hmm. do all mathematicians agree -0 = +0? (possibly a stupid question)
But back to the topic: maybe in this context it means x doesn't exist.
Yes, +0 = -0. You could probably come up with a system where that wasn't true, but then zero would not be the additive identity so calling it "zero" would be misleading.
I don't know what's supposed to be so interesting about the statement x = not x. It's just a false statement, like 1 = 2.
Tom Mattson
Jun12-04, 10:13 PM
x is not_x in an excluded-middle reasoning system, is the reason why we are calling it a paradox.
I hate to sound like Bill Clinton, but what do you mean by "is"? Do you mean material equivalence? If so, then "x is ~x" is not a paradox. It's just false.
Tom Mattson,
'is' equal to '='.
Please read again #1. It is clearly written there.
Tom Mattson
Jun13-04, 02:29 AM
Tom Mattson,
'is' equal to '='.
Please read again #1. It is clearly written there.
OK, in that case: "x is ~x" is not a paradox. It is a false statement.
Russell E. Rierson
Jun13-04, 05:29 AM
By tautology x = x means: x is itself, otherwise we cannot talk about x.
Now we can ask if a teotology is also recursive, for example: x = x = x = ...
If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.
So, Russell's paradox is like if by teotology we ask if x is not_x or x = not_x , which is a meaningless question trough an exluded-middle point of view.
Because our one_step_recursion cannot be done, we have no product of tautology and we can conclude that the whole idea of Russell's paradox simply does not exist in our excluded-middle logical reasoning system.
By the way, in Russell's paradox x is not the set, but the word "include".
So I hope that you agree with me that the question: Is "include = "not include", is meaningless through an excluded-middle reasoning.
What do you think?
The set of all sets that are not members of themselves. Seems to boil down to incomplete definitions? Insufficiency of language? exclusion/inclusion ?
So the set of all dogs is not a member of itself, since, it is not a dog.
But the "dog" identity, is an abstract Platonic form, that gives the aspect of "dogness" to all dogs. The identity is self contained.
So a generalization of the set axioms certainly would help, and we can stop flogging the tired ol' ZF horse. Ergo, a merger of symmetry and ZF theory is of paramount importance.
Hi Russell E. Rierson,
Please read this again:
By tautology x = x means: x is itself, otherwise we cannot talk about x.
Now we can ask if a teotology is also recursive, for example: x = x = x = ...
If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.
So, Russel's paradox is like if by teotology we ask if x is not_x or x = not_x , which is a meaningless question trough an exluded-middle point of view.
Because our one_step_recursion cannot be done, we have no product of tautology and we can conclude that the whole idea of Russel's paradox simply does not hold in an excluded-middle logical reasoning.
By the way, in Russell's paradox x is not the set, but the word "contain".
So I hope that you agree with me that the question: Is "contain" = "do_not_contain", is meaningless through an excluded-middle reasoning.
The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.
The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
(remark: the existence of a set is not dependent on its content for example:
The empty set exists as a framework which examines the abstract idea of "emptiness".
In short, only the name of the set depends on its property, but not its own existence as a framework.)
Therefore there is nothing here that can be found as a state of a paradox.
What do you think?
master_coda
Jun13-04, 12:14 PM
So I hope that you agree with me that the question: Is "contain" = "do_not_contain", is meaningless through an excluded-middle reasoning.
The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.
The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
(remark: the existence of a set is not dependent on its content for example:
The empty set exists as a framework which examines the abstract idea of "emptiness".
In short, only the name of the set depends on its property, but not its own existence as a framework.)
Therefore there is nothing here that can be found as a state of a paradox.
What do you think?
Do you not understand the difference between a false statement and a meaningless one? "contain" = "do_not_contain" is not a meaningless statement, it is a false one.
And the problem of the set of all sets that do not contain themselves is not one that can be solved by waving your hands and saying "it doesn't contain itself because this is its self identity". The problem is that if the set exists then you can show that the set contains itself if and only if the set does not contain itself, which contradicts the law of the excluded middle.
You can only work around this problem by doing one of two things:
1) You can use a system of logic that does not include the law of the excluded middle. This significantly weakens your system of logic by making the fact that a statement is true almost meaningless.
2) You can use a version of set theory that does consider the set of all sets that do not contain themselves to be a set. This is the approach ZF set theory takes. It avoids the paradox by eliminating certain types of sets.
The problem is that if the set exists ...
You do not understand my tautology/recursion argument.
Nothing can exist and also contrarict its own existence, therefore Russell's Paradox simply deos not exist and there is no "if" here.
You can wave with your "if" as much as you want, and still Russell's Paradoxs is meaningless exactly like these meaningless questions:
"Is a layer is a honest?" or "Is a honest is a layer?"
"Is black is white?" or "Is white is black?"
There is no false here but only meaningless questions.
"it doesn't contain itself because this is its self identity" and there is no "if" here!
Also you missed the most importent point which is:
Not the set is examined here but the meaningless question:
Is "contain" is "do_not_contain?"
master_coda
Jun13-04, 02:19 PM
You did not understand my tautology/recursion argument.
Nothing can exist and also contrarict its own existence, therefore Russell's Paradox simply deos not exist and there is no "if" here.
You can wave with your "if" as much as you want, and still Russell's Paradoxs is meaningless exactly like these meaningless questions:
"Is a layer is a honest?" or "Is a honest is a layer?"
"Is black is white?" or "Is white is black?"
There is no false here but only meaningless questions.
Russel's Paradox does not contradict its own existance, so your argument is invalid. The paradox is just that the existance of the set of all sets that do not contain themselves contradicts the law of the excluded middle. Thus any version of set theory that states that such a set exists is inconsistent.
Your examples you meaningless questions (they actually aren't even questions, not even meaningless ones) have nothing to do with Russel's paradox.
Also you missed the most important point which is:
Not the set is examined here, but the meaningless question:
Is "contain" is "does_not_contain?"
The set concept is a natural concept here and we can change it by anything that can or cannot contain things.
master_coda
Jun13-04, 02:34 PM
Also you missed the most important point which is:
Not the set is examined here, but the meaningless question:
Is "contain" is "does_not_contain?"
The set concept is a natural concept here and we can change it by anything that can or cannot contain things.
But this has nothing to do with Russel's paradox. It's just a meaningless question because it makes no grammatical sense.
Unless you mean "is contain = does_not_contain" which is not meaningless either. The answer is just NO. Much like "is true = false" can also be answered NO.
It is connected to Russell's paradox.
The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.
The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
Now please omit the green and red words from the above sentences and you have lost the argument here.
It means that the whole "story" here is around the green and the red words.
green word cannot be but a green word.
red word cannot be but a red word.
You can add the set concept or not, but it does not change this tautology/recursion argument.
Therefore Russell's Paradox does not exist.
If by your reasoning "does not exist" = "false", then Russell's Paradox is false.
master_coda
Jun13-04, 03:04 PM
The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
But if the set does not contain itself, then it must be contained in the set of all sets that do not contain themselves. Thus the set must contain itself.
Then you ignore again the self identity of x to itself.
master_coda
Jun13-04, 03:13 PM
Then you ignore again the self identity of x to itself
I don't have to ignore anything. No matter how many defintions you add, and how many different ways you can prove that the set does not contain itself, the contradiction I demonstrated will always exist. Any logical system that is inconsistent cannot be fixed by adding more to the system.
Maybe this analogy can help.
Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.
Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.
Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recusion does_not_contain.
It is clear that they cannot be in each other states without first to lose their own existence (self identity).
The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself, because this is its self identity.
The set of ALL_sets_that_ do_not_contain_themselves,
must not contain itself as a member of itself, because this is its self identity.
This is not "self-identity"; this is the "fallacy of composition". In general, the whole does not have the properties of its parts.
"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.
Maybe this analogy can help.
Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.
Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.
Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recusion does_not_contain.
It is clear that they cannot be in each other states without first to lose their own existence (self identity).
master_coda
Jun13-04, 03:36 PM
Maybe this analogy can help.
Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.
Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.
It would be nice if when you used words like tautology, recursion and fractal you used them with the meanings everyone else uses and not your own, but that isn't really a problem here.
If you use the standard ideas of what a set is and what "contains" is in naive set theory, then Russel's paradox is in fact a paradox. If you change the definitions of "set" and "contains" to something else, then it's possible that Russel's paradox is not a problem with the new definitions. However that does not alter the fact that Russel's paradox is a paradox in naive set theory...it just avoids the problem by using a different kind of set theory.
Sorry Master_coda,
But again you miss the fine point of my previous post, whit is the word "first".
So here is my analogy again and this time pay attention to this word:
Because by the tautology/recursion idea the tautology x = x is like the recursion x = x = x = ... then we can think on a fractal-like information form.
Any part of our fractal is like the whole fractal, and if we have a green fractal it cannot be with any self identity relation with, for example, a red fractal, and vise versa.
Now let us say that a green fractal is the tautology/recusion contain and a red fractal is the tautology/recursion does_not_contain.
It is clear that they cannot be in each other states without first to lose their own existence (self identity).
"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.
And also "self-identity" say that the set of all_sets_that_do_not_contain themselves must be the set of all_sets_that_do_not_contain themselve.
master_coda
Jun13-04, 04:05 PM
Sorry Master_coda,
But again you miss the fine point of my previous post, whit is the word "first".
And you continue to miss the point. This is an entirely new system you've invented, so it has nothing to do with any existing theory. So it proves nothing about how Russel's paradox applies to other theories. The "does not contain" that you are refering to is not the "does not contain" relation used in the construction of Russel's paradox.
No set can be its opposite ("contain" , "does_not_contain") without first losing its own existene.
Therefore the circular state of Russell's Paradox does not exist.
master_coda
Jun13-04, 04:46 PM
No set can be its opposite ("contain" , "does_not_contain") without first losing its own existene.
Therefore the circular state of Russell's Paradox does not exist.
So your argument is that Russel's paradox does not exist because if it did the set that contains all sets that do not contain themselves would not exist?
Normally when you have axioms and you derive a contradiction from those axioms, you fix the problem by changing your axioms, not stating that logic does not exist.
Russell E. Rierson
Jun14-04, 12:33 AM
This is not "self-identity"; this is the "fallacy of composition". In general, the whole does not have the properties of its parts.
"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.
The correct logic is:
If A then B
A
therefore B
Yet the fallacy of composition is equivalent to:
If A then B
B
therefore A
which is incorrect logic.
The set of natural numbers has the identity "natural number" that distributes over all members of the "set"[the whole distributes over the parts].
The most fundamental identity distributes over all elements of the "Universal Set". True, one specific aspect is not a universal property but the universal property can be the first step in the logical deduction that eventually leads to the specific aspect.
U[X[Y[Z...{ }]]]
Russell's paradox is a form of the liars paradox:
This statement is false
Which leads to Goedel's incompleteness theorem.
No set can be its opposite ("contain" , "does_not_contain") without first losing its own existence.
Therefore the circular state of Russell's Paradox does not exist.
Now, Let us go deeper then that:
No set can be its opposite ("contain" , "does_not_contain") without first losing its own identity.
Therefore the circular state of Russell's Paradox cannot be found.
There is a very deep idea here that can be used as the basis of what I call "A non-naive Mathematics".
By a non-naive Mathematics the existence of an element does not depend on its name.
For example, let us take two different points.
The existence of the points is not depending on their names.
It means that the two points can have any pair of different names.
Now, let us say that names are what we call numbers.
So each number, when mapped with some point, give it its unique identity.
We get here two basic systems:
The absolute system:
Made of infinitely many points, which their existence does not depend on their identity (which is some unique name that can be mapped to each one of them).
The relative system:
Made of infinitely many possible unique names that when mapped with some absolute point, they determinate its identity.
It means that the identity of any absolute point relatively can be changed by the current name that we give it (after two arbitrary and unique names are given, the rest of points/names mapping is well-defined, relatively to an arbitrary name, which is used as a global name of the entire points/names mapping).
In the case of numbers, the global name is actually a unique scale factor over
the entire real-line (for more detailes about the real line, please look at http://www.physicsforums.com/showthread.php?t=30254)
This interaction between absolute/relative concepts, is maybe the deepest foundation of the language of Mathematics and can be used a solid basis to define its organic dynamical structure.
"There does not exist an x such that x is not equal to x" is a perfectly correct statement. This does not mean that we aren't allowed to write "x != x"; it simply means that this statement is false.
And Russel, the fallacy of composition is not equivalent to what you wrote.
Hi Hurkyl,
Please read carefully post #37 (all of it, including the link) thank you.
master_coda
Jun14-04, 07:23 AM
Hi Hurkyl,
Please read carefully post #37 (all of it, including the link) thank you.
Perhaps you should start reading our posts.
I read yours, and this is the reason why I came up with a new theory of a non-naive-mathematics.
No, it is simpler than that. :wink:
By a non-naive Mathematics the existence of an element does not depend on its name.
master_coda
Jun14-04, 01:08 PM
No, it is simpler than that. :wink:
By a non-naive Mathematics the existence of an element does not depend on its name.
Apparently you don't realize that in actual math, the existance of something does not depend on its name either. Naming has absolutely nothing to do with Russel's paradox, although prehaps your lack of understanding has led you to believe that it does.
http://plato.stanford.edu/entries/russell-paradox/Accordingly the solution to the paradox can be found in the key word "naive" set theory.Obviously, some classes are not well defined sets that obey logical operations.In other words,they are not correctly defined.
Axiomatic set theories are required to prevent paradoxes.
Apparently you don't realize that in actual math, the existance of something does not depend on its name either. Naming has absolutely nothing to do with Russel's paradox, although prehaps your lack of understanding has led you to believe that it does.
No, master_coda you are the one how misunderstand the meaning of identity.
If A is "The set of all_sets_that_do_not_contain_themselves" then its own self identity is not to contain itself.
If B is "The set of all_sets_that_do_contain_themselves" then its own self identity is to contain itself.
Now, A self identity cannot be related to any Blue(=B) property.
Also, B self identity cannot be related to any Red(=A) property.
Russell's paradox is based on this state:
,---> self identity B ---, | |
Self identity A is observed as | |
| |
'--- self identity A <---'
Now, we have to understand that there is no way to observe identity A as if it has also properties of identity B and also to keep its A identity on the same time.
The "paradox" arises because we rape by force identity A to keep its own identity and also to say that it has a B property.
Conclusion:
Russell's Paradox is nothing but a brutal action of a rough mind.
matt grime
Jun15-04, 03:11 AM
you'd do a lot better i f you didn't use such silly terms as 'rape' and 'brutal' and 'ruff' (which is a frilly neck garment by the way) and actually learned about the stuff you're pronouncing on. Russell's paradox simply arises from the naive attempt to say that a set is a collection of objects with a rule for belonging to that set.
Russell E. Rierson
Jun15-04, 03:32 AM
And Russel, the fallacy of composition is not equivalent to what you wrote.
Wake up and smell the logic Hurkyl. It is a MAD world!
The composition fallacy is a form of "modus ponens" error:
http://www.illc.uva.nl/j50/contribs/eemeren/eemeren.pdf
The fallacies of composition and division
Frans H. van Eemeren, University of Amsterdam and New York University
Rob Grootendorst, University of Amsterdam
1. Introduction
In the pragma-dialectical conception of argumentation fallacies are defined as violations of rules that further the resolution of differences of opinion. Viewed within this perspective, they are wrong moves in a discussion. Such moves can occur in every stage of the resolution process and they can be made by both parties. Among the wrong moves that can be made in the argumentation stage are the fallacies of composition and division. They are violations of the rule for reasonable discussions that any argument
used in the argumentation should be valid or capable of being validated by making explicit one or more unexpressed premises. In this paper the fallacies of composition and division are analyzed in such a way that it becomes clear that the problem at stake here is in fact a specific problem of language use.
2. Properties of wholes and the constituent parts
There are several ways of violating the dialectical rule that the reasoning that is used in argumentation should be valid or capable of being validated by making explicit one or more unexpressed premises. To make this clear, first, the argument has to be reconstructed that is used in the argumentation. Next, an intersubjective reasoning procedure has to be gone through to establish whether the argument is indeed valid (van
Eemeren and Grootendorst 1984: 169).
A well-known violation of the validity rule consists of confusing necessary and
sufficient conditions in reasoning with an 'If ... then' proposition as a premise.
There are two variants. The first is the fallacy of affirming the consequens, in which, by way of a 'reversal' of the valid argument form of modus ponens, from the affirmation of the consequens (by another premise) is derived that the antecedens may be considered confirmed. The second is the fallacy of denying the antecedens, in which by way of a similar reversal of the valid argument form of modus tollens the denial of the consequence is derived from the denial (by another premise) of the antecedens.
There are also other violations of the validity rule. A violation that often occurs is unjustifiably assigning a property of a whole to the constituent parts. Or the other way around: unjustifiably assigning a property of the constituent parts to the whole. The properties of wholes and of parts are not always just like that transferable to each other. Sometimes the transfer leads to invalid reasoning:
a This chair is heavy
b Therefore: The lining of this chair is heavy
Lama :
You have a nice name in Hebrew mean "Way".
Well I see that you treat symbol as mathematical object and by these Russell paradox have a new meaning. Please tell me and how is all that relate if at all to the Epilog of the book "Nature's number" by Ian Stewart and his interesting new idea about Morfomatica?
Thank you
Moshek
:shy:
You have a nice name in Hebrew mean "Way".
"Lama" in Hebrew is "Why?" and not "Way".
you'd do a lot better i f you didn't use such silly terms as 'rape' and 'brutal' and 'ruff' (which is a frilly neck garment by the way) and actually learned about the stuff you're pronouncing on. Russell's paradox simply arises from the naive attempt to say that a set is a collection of objects with a rule for belonging to that set.
Thank you, I corrected "ruff" to "rough".
I used 'rape' and 'brutal' and 'rough' not as mathematical terms but to clearly show how some fundamental parts of Modern Mathematics do not hold water.
No, master_coda you are the one how misunderstand the meaning of identity.
No, I must insist that you are the one misunderstanding identity.
Self-identity says "A thing is equal to itself", which is something vastly different than the fallacy of composition, which says "A thing satisfies the properties of its parts".
Some other examples:
The "set of all individual numbers" is clearly not an individual number.
The "set of everything worth less than a dollar" would obviously be worth more than a dollar.
This "set of all blue objects" is clearly a red object.
Wake up and smell the logic Hurkyl. It is a MAD world!
The composition fallacy is a form of "modus ponens" error:
Your quote from that link seems to say exactly the opposite...
A thing satisfies the properties of its parts
In this case A cannot be in a state that do not satisfies the properties of its parts, and this it exactly what I say.
The set of all dogs is not a dog itself but it has a common property with each dog that included in it, which we can call it "Dogness".
So the most basic identity of each dog and the most basic identity of the set of all dogs is "Dogness" (in this case this basic identity is also like a one_step_recursion, which is equivalence to the tautology x=x).
Now in the case of Russell's Paradox, the most basic identity of each member "not_to_contain_itself" (which is like the "Dogness" example) and the most basic identity of the set of "all_members_that_do_not_contain_themselves" is "not_to_contain_itself" (in this case this basic identity is also like a one_step_recursion, which is equivalent to the tautology x=x).
Strictly speaking, the "Dogness" identity example is equivalent to the "not_to_contain_itself" identity case.
The "set of everything worth less than a dollar" would obviously be worth more than a dollar.
In this case it cannot be a member of itself because the most basic identity here is "worth less than a dollar".
IN EACH CASE WE HAVE TO DEFINE THE MOST BASIC PROPERTY, AND ONLY THEN WE CAN CONCLUDE IF THIS PROPERTY MEANS THAT WE HAVE TO INCDLUDE THE SET IN ITSELF.
FOR EXAMPLE: THE SET OF ALL_MEMBERS_THAT_CONTAIN_THEMSELVES MUST CONTAIN ITSELF AS A MEMBER OF ITSELF, BECAUSE “TO_CONTAIN_YOURSELF” IS THE MOST BASIC IDENTITY IN THIS CASE.
IN SHORT, RUSSELL'S PARADOX DOES NOT HOLD WATER JUST BECAUSE OF THE REASON THAT THERE IS NO LOGIC STATE HERE THAT FORCE US TO INCLUDE THE SET IN ITSELF.
If A is "The set of all_sets_that_do_not_contain_themselves" then its own self identity is not to contain itself.
If B is "The set of all_sets_that_do_contain_themselves" then its own self identity is to contain itself.
Now, A self identity cannot be related to any Blue(=B) property.
Also, B self identity cannot be related to any Red(=A) property.
Russell's paradox is based on this state:
,---> self identity B ---, | |
Self identity A is observed as | |
| |
'--- self identity A <---'
Now, we have to understand that there is no way to observe identity A as if it has also properties of identity B and also to keep its A identity on the same time.
The "paradox" arises because we force identity A to keep its own identity and also to say that it has a B property.
matt grime
Jun15-04, 09:32 AM
you should stop using your 'real life' intuition in mathematics, Doron, in particular your notion of 'sharing' some element of 'dogness' which is a spurious example to do with your subjective notion of degree.
Hi Matt,
Please refreash your screen and read all of my previous post, thank you.
matt grime
Jun15-04, 10:03 AM
but the set of dogs displays no aspect of 'dogness' ie being a dog. its elements do. learn, please, before spouting asinine garbage.
The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.
"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".
"DOGNESS" IS NOT ANY PARTICULAR DOG.
By the way, to say that a set is not its members is a non-abstract and trivial thing to say.
If you build your abstarct logical reasoning on this non-abstract trivial thing, then you can't go far.
master_coda
Jun15-04, 12:24 PM
By the way, to say that a set is not its members is a non-abstract and trivial thing to say.
If you build your abstarct logical reasoning on this non-abstract trivial thing, then you can't go far.
Of course, your idea of basing reasoning on abstract, contradictory things is far better.
Understand this simple fact - you cannot assume that a set must share any properties with the things it contains. If you do make this assumption, one of two things will happen:
1) your system of logic will be inconsistent
2) your system of logic will be unable to make any deductions about sets that do not share any properties with their elements, and will therefore be much, much weaker than existing logical systems
matt grime
Jun15-04, 12:44 PM
The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.
"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".
"DOGNESS" IS NOT ANY PARTICULAR DOG.
I realize this isn't being conducted in your first language, but at least take care to read what is written. you are saying that the set of dogs displays the properties of being 'doggy'', and that is certainly not true. i did not say the set of dogs has nothing to do with dogs. that would require some agreement on what we mean by 'has to do with'.
Consider the set which contains the empty set, that set is not empty...
Russell E. Rierson
Jun15-04, 12:47 PM
The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.
"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".
"DOGNESS" IS NOT ANY PARTICULAR DOG.
Perhaps "dogness" can be viewed as a form of constraint forcing the members included in the set of dogs to this defining aspect.
The set of all dogs is a subset of the set of all mammals...
Eventually, the set that includes "everything" is reached by removing nested constraints.
Russell E. Rierson
Jun15-04, 01:12 PM
Your quote from that link seems to say exactly the opposite...
Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?
Consider the set which contains the empty set, that set is not empty...
A set is only a framework where we can examine our ideas, and its own existence does not depend on the properties of its contents.
Only its name (identity) is denpend on the properties of its contents.
Again you use a non-abstract approech of the set concept.
As for "Dogness", I use this world as the most geneal concept of anything that is realed to dogs, but also does not have to be a dog at all.
If you have another word instead of "Dogness" to what I wrote above, then I'll be glad to get it from you.
Understand this simple fact - you cannot assume that a set must share any properties with the things it contains. If you do make this assumption, one of two things will happen:
1) your system of logic will be inconsistent
2) your system of logic will be unable to make any deductions about sets that do not share any properties with their elements, and will therefore be much, much weaker than existing logical systems
1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.
2) Please show us some mathematics where sets with no names (identities) are involved.
master_coda
Jun15-04, 01:57 PM
1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.
2) Please show us some mathematic where sets with no names (identiteis) are involved.
What does that have to do with anything? Of course you need to know about the properties of the contents of the set. My point was that just because a property holds true for every element in a set, you cannot then conclude that the property also holds true for the set itself.
matt grime
Jun15-04, 02:18 PM
perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.
you cannot then conclude that the property also holds true for the set itself.
Identity of a set has nothing to do with 'true' or 'false'.
The identity of a set is based on the most abstract basis of its contents, which gives it its name.
Therefore no set can contrarict its own identity (name).
In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.
This is exactly as if we say: Black is White or vise vera.
perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.
Please read post #63
master_coda
Jun15-04, 03:26 PM
Identity of a set has nothing to do with 'true' or 'false'.
The identity of a set is based on the most abstract basis of its contents, which gives it its name.
Therefore no set can contrarict its own identity (name).
In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.
This is exactly as if we say: Black is White or vise vera.
"No set can contradict its own name" isn't just something you can just assert. It doesn't even make sense. Your "proof" is nothing more than you saying "I made up a rule about sets, and Russel's paradox violates it, so the paradox must be wrong".
What are you going to do next? Tell us that 0 = 1 and so obviously x/0 = x?
Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict its self identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?
Can you be master-coda which is not master-coda?
master_coda
Jun15-04, 04:01 PM
Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict its self identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?
Can you be master-coda which is not master-coda?
If you have A = "set of all sets that contain themselves" then all you know is that if a set is in A, then that set must contain itself. The definition does not say that A must also contain itself.
If B = "set of all even numbers" you cannot assume that B must itself be an even number. If C = "set of all sets that contain themselves" you cannot assume that C is a set that contains itself. If D = "set of all sets that do not contain themselves" then you cannot assume that D must not contain itself.
And, even if you were to add an axiom to your system that said "the set of all sets that do not contain themselves does not contain itself", Russel's paradox still applies. No amount of whining that the set must not contain itself will change that.
In this case A cannot be in a state that do not satisfies the properties of its parts, and this it exactly what I say.
Then you have to prove it. You have to prove that A satisfies the condition required to be a member of A.
The set of all dogs is not a dog itself but it has a common property with each dog that included in it, which we can call it "Dogness".
Can you show one example of "dogness" that is shared by "the set of all dogs"?
1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.
"The set of all dogs" gets is name precisely because:
(1) every dog is contained in the "set of all dogs"
(2) everything in the "set of all dogs" is a dog
There is no reason to think that the "set of all dogs" should have any properties of dogs. The relation here is:
A thing is a member of the "set of all dogs" if and only if that thing is a dog.
But, finally, "the set of all dogs" doesn't need to have a name; it can be discerned completely by the two properties I listed above.
In fact, pay attention to the fact that the name "the set of all dogs" is not really a name at all; it is a phrase stating what objects are in the set!
Speaking loosely, the "identity" of a set is entirely determined by the identities of its contents. This is stated in the axioms of ZFC by "A and B are the same set if and only they have the same elements".
By the way, to say that a set is not its members is a non-abstract and trivial thing to say.
You have this exactly backwards. A set is an abstract thing, whose "identity" is given entirely by the concept of "membership". The fact that you are unwilling (unable?) to seperate the two ideas "properties of a set" and "properties of the elements of a set" is a very strong indicator of non-abstract thinking.
Master coda, think simple.
If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.
And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.
master_coda
Jun15-04, 04:52 PM
Master coda, think simple.
If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.
And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.
Again, these are nothing more than rules you have just made up.
Hurkyl,
From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.
But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.
In short, please read ,for example, post #63.
master_coda
Jun15-04, 05:02 PM
Hurkyl,
From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.
But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.
In short, please read ,for example, post #63.
Ha! You critizing people for lack of understanding is just funny. You don't even know the basic principles of logic, you don't know how to define things, and you can barely produce coherent English.
Perhaps you should consider the possibility that we all haven't fallen over ourselves worshipping your brilliance because you ideas make no sense, and not because we are stupid.
Again, these are nothing more than rules you have just made up.
No you use these simple rules to keep the identity of somthing.
These are the most simple rules that for example keeping you for not be me. :wink:
and you can barely produce coherent English.
My language is Hebrew.
..and not because we are stupid.
I think that Hurkyl, Matt Grime and you dear master coda are very intelegent and wise persons.
I simply have another point of view on the foundations of the language of Mathematics and to what directions it has to be developed.
Self-identity says "a thing is itself"; nothing more, nothing less.
In the case where we're considering the thing "The set of all sets that contain themselves", self identity says:
"The set of all sets that contain themselves" is "the set of all sets that contain themselves".
Notice that this is a different statement than
"The set of all sets that contain themselves" is a "set that contains itself".
The other examples I mentioned are intended to show this. The following three statements are all false:
"The set of all dogs" is a "dog"
"The set of everything worth less than a dollar" is "worth less than a dollar"
The set of everything blue is blue.
The point is, "self-identity" cannot be used (by itself) to prove:
"The set of all T" is a "T".
Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?
Any example would do.
Here's one: \forall x \in A: P(x) therefore P(A).
The point is, "self-identity" cannot be used (by itself) to prove:
Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict its self identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?
Can you be Hurkyl which is not Hurkyl?
"The set of all sets that contain themselves"
A set with no identity is only the green part.
Some identity (name) is the red part.
We need both green and red parts to make Math.
And the red part cannot contradict itself during Math operations.
The problem is that you're not being consistent about when you use the green part.
Self identity says:
The set of all sets which contain themselves is The set of all sets which contain themselves.
Furthermore, we have the tautology
A set which contains itself is a set which contains itself.
However, you are saying
The set of all sets which contain themselves is a set which contains itself.
You can either have the green on both sides of "is", or on neither side; it is (usually) incorrect to mix and match.
Sorry Hurkyl,
But you have changed what I wrote.
The green part is: The set of...
The red part is: all_sets_that...
Russell E. Rierson
Jun16-04, 05:42 AM
Any example would do.
Here's one: \forall x \in A: P(x) therefore P(A).
If A then B
If each stick is breakable, then the whole bundle of sticks is breakable.
The fallacy of composition has a faulty premise.
If each stick is breakable, then the whole bundle of sticks is breakable.
It is like a self similarity of a fractal.
What I suggest is very simple:
1) The language of Mathematics is based on two systems: The relative and the absolute.
2) The absolute system is a finite or infinitely many elements with no unique identity.
3) The relative system is a finite or infinitely many possible unique names.
4) If a possible unique name is related to some absolute element, it determines its identity.
5) There are two basic types of operations on an element with a unique identity:
a) An operation that changes its identity.
b) An operation that does not change its identity.
6) In an excluded-middle reasoning, an absolute element can have simultaneously a one and only one unique name (identity).
If the absolute element is a set under an excluded-middle reasoning, then:
1) Its identity depends on the most abstract property of its content; therefore it cannot contradict the most abstract property of the content.
2) This identity cannot be changed under any operation, unless the most abstract property of the content is changed.
3) If the identity of a set is changed under an operation, its previous identity is not related to it anymore.
master_coda
Jun16-04, 07:45 AM
Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.
Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.
An identity of a set cannot be but with a relation with the most abstract proprty of the content.
It is very fundamental.
By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.
The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Language of Mathematics.
For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
master_coda
Jun16-04, 01:56 PM
An identity of a set cannot be but with a relation with the most abstract proprty of the content.
It is very fundamental.
By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.
The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Language of Mathematics.
For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
You seem to be treating abstact ideas as if they are phsyical objects that can be changed or destroyed.
For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.
A set which exists cannot be somehow made to not exist. You can construct a new set that has weaker properties (the equivalent of "removing" the properties or "identity" of the original set) but that does not make the original set no longer exist.
matt grime
Jun16-04, 06:21 PM
none of the things you suggest mathematics must be, or do, is informed, lama/doron/shmesh/etc, and just demonstrates your complete lack of understanding, and you total ignorance of the world you claim to talk about. Are you even aware of topoi where the 'excluded middle' fails to be true? No, you aren't. Mathematics is far richer than you can even begin to understand, and the repeated demonstrations of your ignorance of it are not particulalry endearining.
You are also inconsitent in the extreme. One need only look at you belief in dichotomic options to see that.
none of the things you suggest mathematics must be, or do, is informed, lama/doron/shmesh/etc, and just demonstrates your complete lack of understanding, and you total ignorance of the world you claim to talk about. Are you even aware of topoi where the 'excluded middle' fails to be true? No, you aren't. Mathematics is far richer than you can even begin to understand, and the repeated demonstrations of your ignorance of it are not particulalry endearining.
You are also inconsitent in the extreme. One need only look at you belief in dichotomic options to see that.
There is a little problem here dear Matt.
You did not show us that you understand my ideas about Math.
Please also read my answer to master coda.
For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.
So, you do not understand the idea of the relative/absolute system.
In an excluded-middle reasoning an absolute element (set, point, ...) can have simultaneously a one and only one unique name (identity).
And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.
The identity of an absolute element is its literal name like: point named 'pi', point named 'e',
point named '1', point named '0', the set of 'all_sets_that_do_not_contain_themselves' ... and so on.
So nothing disappears here.
A set which its literal name is 'all_sets_that_do_not_contain_themselves' cannot contain itself (in an excluded middle reasoning) exactly as some absolute point cannot have more than one literal name (in an excluded-middle reasoning).
Conclusion: Russel's Paradox cannot be defined in an excluded-middle reasoning.
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.
I am waiting for your detailed remarks of my papers.
Thank you,
Lama
Russell E. Rierson
Jun17-04, 01:25 AM
What I suggest is very simple:
The language of Mathematics...
A description is an abstract representation of a concrete physical instantiation.
Mathematics is a meta-language, highly abstract. A description contains the concrete physical instantiation in the abstract sense and the concrete object contains the description in the physical sense.
Here is the definition of "algorithm":
http://en.wikipedia.org/wiki/Algorithm
"Algorithm
From Wikipedia, the free encyclopedia.
Broadly-defined, an algorithm is an interpretable, finite set of instructions for dealing with contingencies and accomplishing some task which can be anything that has a recognizable end-state, end-point, or result for all inputs. (contrast with heuristic). Algorithms often have steps that repeat (iterate) or require decisions (logic and comparison) until the task is completed."
DNA is an algorithm, a finite set of instructions, which can construct a carbon based life form.
The life form physically contains the DNA and the DNA contains the life form in an "abstract" sense.
At a fundamental level of existence, it is postulated that "nature" could be constructed of tiny strings, and those strings, loops, or branes, could even be constructed of string "bits".
These bits could encode information, analogous to the universe's "DNA"? A set of instructions built into the fabric of space/time and mass/energy?
"If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. I hold it true that pure thought can grasp reality, as the ancients dreamed." (Albert Einstein, 1954)
At the most fundamental length scales, the fundamental paticles, called "strings", could be constructed of even more basic units i.e. bits? analogous to a computer code?
1010100010...etc.
Universal algorithms?
Some interesting ideas on "string bits":
http://xxx.lanl.gov/PS_cache/hep-th/pdf/9607/9607183.pdf
http://xxx.lanl.gov/PS_cache/hep-th/pdf/9707/9707048.pdf
Introduction
In string-bit models, string is viewed as a polymer molecule, a bound system of point-like constituents which enjoy a Galilei invariant dynamics. This can be consistent with Poincar´e invariant string, because the Galilei invariance of string-bit dynamics is precisely that of the transverse space of light-cone quantization. If the string-bit description of string is correct, ordinary nonrelativistic many-body quantum mechanics is the appropriate framework for string dynamics. Of course, for superstring-bits, this quantum mechanics must be made supersymmetric.
According to string theory, the uncertainty in position is given by:
Dx < h/Dp + C*Dp
Which points towards a type of "discrete" spacetime?
A metric space has distance function r(x,y), characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles.
If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.
According to the Pythagorean theorem:
x^2 + y^2 = z^2
All possible integer solutions are then rerpresented as:
[a^2 - b^2]^2 + [2ab]^2 = [a^2 + b^2]^2
a^4 -2(ab)^2 + b^4 + 4(ab)^2 =
a^4 + 2(ab)^2 + b^4 = [a^2 + b^2]^2
all odd numbers can be represented as:
[a^2 - b^2] or Z^p - Y^p
if Y is an "even" natural n and Z is odd, same for a and b .
Fermat's last theorem, for integers a,b,Z,Y,p:
[a^2 - b^2]^p + Y^p = Z^p
[a^2 - b^2]^p = Z^p - Y^p
a^2 - b^2 = [Z^p - Y^p]^[1/p]
When Z^p - Y^p is a prime number, it cannot have an integer root.
a^2 - b^2 is not an integer, for [Z^p - Y^p]^[1/p] , for a,b,Z,Y,p, unless p = 2.
To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything. [o)]
This means that relativity holds in the "topological" sense and T-duality is correct.
Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".
The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.
[density 1]--->[density 2]--->[density 3]---> ... --->[density n]
[<-[->[<-[->[U]<-]->]<-]->]
Intersecting wavefronts = increasing density of spacelike slices
As the wavefronts intersect, it becomes a mathematical computation:
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n
For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.
So, you do not understand the idea of the relative/absolute system.
In an excluded-middle reasoning an absolute element (set, point, ...) can have simultaneously a one and only one unique name (identity).
And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.
The identity of an absolute element is its literal name like: point named 'pi', point named 'e',
point named '1', point named '0', the set of 'all_sets_that_do_not_contain_themselves' ... and so on.
So nothing disappears here.
A set which its literal name is 'all_sets_that_do_not_contain_themselves' cannot contain itself (in an excluded middle reasoning) exactly as some absolute point that cannot have more than one literal name (in an excluded-middle reasoning).
Conclusion: Russel's Paradox cannot be defined in an excluded-middle reasoning.
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.
I am waiting for your detailed remarks of my papers.
Thank you,
Lama
Dear Russell E. Rierson,
Can you simply say how you connect my idea of relative/absolute system to your interesting ideas?
Russell E. Rierson
Jun17-04, 01:56 AM
Dear Russell E. Rierson,
Can you simply say how you connect my idea of relative/absolute system to your interesting ideas?
In a word, "duality".
In a word, "duality".
What is "duality" for you?
Russell E. Rierson
Jun17-04, 02:42 AM
What is "duality" for you?
The laws of physics become the laws of geometry. Certain invariants hold, which are analogous to the "absolutes". There are also analogous non-absolutes, or relational perspectives, on the surface of the geometry.
Russell E. Rierson
Jun17-04, 02:55 AM
What is "duality" for you?
Here is an article on symmetry and duality. It appears to be very interesting:
http://1omega.port5.com/articles/Sym_dual/SYM_DUAL.HTM
Dear Russell E. Rierson,
It is about the time to build our community, which is based on the "Duality" principle.
I have learned during the last 20 years that without a community support, no fundamentals can be changed in science.
Do you have any ideas?
Russell E. Rierson
Jun17-04, 05:15 AM
Dear Russell E. Rierson,
It is about the time to build our community, which is based on the "Duality" principle.
I have learned during the last 20 years that without a community support, no fundamentals can be changed in science.
Do you have any ideas?
Duality, or possibly, a three valued logic as phoenix explains, solves russell's paradox
Russell E. Rierson,
"Duality" principle is not necessarily a 2-valuad logic, when you examine it by an Included-Middle Logic.
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.
I am waiting for your detailed remarks of my papers.
Thank you,
Lama
The simple principle of Duality is an Archimedean point
That will make a shift in the whole mathematics.
Moshek
Russell E. Rierson
Jun18-04, 02:40 AM
Russell E. Rierson,
"Duality" principle is not necessarily a 2-valuad logic, when you examine it by an Included-Middle Logic.
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.
Lama
The number of fractions from zero to one, is equal to the number of "natural numbers", from zero to infinity.
The number of fractions from zero to one, is equal to the number of "natural numbers", from zero to infinity.
Please give a detailed explanation.
Russell E. Rierson
Jun18-04, 05:15 AM
Please give a detailed explanation.
In your paper, you wrote that any segment, has the same magnitude as the entire real line.
I was trying to understand :biggrin:
Let us add some more details.
Let us say that every unique name along the real line is represented by a single symbol.
In this case there is 'no room' for Cantor's diagonal method and we cannot conclude that there is a difference between the single symbols of the entire real line and the single symbols of the natural numbers.
But this is not correct because, when we ignore the fractal nature of the real line and care only about its magnitude, then in this case any unique symbol can be mapped only to itself.
In this case, the unique symbols that represent only the natural numbers cannot have the magnitude of the entire real line.
The mistake of standard Math point of view is: when it finds a 1-1 and onto between some set of infinitely many elements to some proper subset of it, it is not aware to the fact that it uses the fractal property of the number line.
If we aware to the simple fact that the magnitude of the number line is not depended on its fractal nature, then and only then we can clearly understand (by researching a one and only one arbitrary level of this fractal) that there cannot be any 1-1 and onto between some set of infinitely many unique symbols, to a proper subset of it.
Strictly speaking, the absolute/relative picture of the real-line is simpler and richer than the standard point of view.
Another important side effect here is that our simple intuitions are not forced to deal with weird states.
You had a mistake when you use the word mistake... for the regular way of thinking in mathematics. I think that what you are trying to do is to show us that there was some blind point and by see this point we can jump to a completely new dimension. I would call it the dimension of the observer.
a fundamental point that most of the physician are so missing by there modeling and equation attitude ( String theory etc.. ) which is still some Newton mathematics in and relatively Einstein universe. Strictly speaking you ( We ?) are talking about not Newtonian mathematics were symbol are object by themselves by the principle of duality. as a positive interpatation to Godel theorem.
Moshek :rofl:
Hi Moshek,
If you examine the meaning of the word 'mistake' you can find within it a combination of two words, which are the words 'miss' and 'take'.
In short, the word 'mistake' and your 'blind point' idea are actually the same.
I disagree with you about your dichotomy point of view of relative and absolute systems.
By my point of view the whole idea of duality is based on the interaction between absolute and relative systems.
Strictly speaking, my system is the interactions between Newton's reasoning and Einstein's reasoning.
In my opinion, no one of them alone can be a meaningful system.
matt grime
Jun18-04, 08:44 AM
If you examine the meaning of the word 'mistake' you can find within it a combination of two words, which are the words 'miss' and 'take'
ACtually, that isn't correct.
Mistake is old norse mis taka, meaning to take wrongly. Miss is dutch and comes from missen, meaning, well, miss
Thank you dear Matt for the correction.
'Mis-' is mostly used as a negative prefix.
Please read also #91 and #106 , thank you.
Hi Doron ,
I don't think that the Euclidian mathematics have mistake in it. But i think that all the great knowledge that was develop make us today the opportunity to develop some new understanding that the logic is not anymore in the center of it. The duality principle is a very good new direction to do that. And your new definition to the concept of number is fundamental and beautiful.
It now only... a matter of creating the new language and community.
Moshe
Thank you very much dear Moshek, I think that both of us are maybe the beginning of this new community.
I am working very hard to find more members.
Yours,
Lama
I edited this thread and it can bo found in: http://www.geocities.com/complementarytheory/Russell1.pdf
Dear Doron:
Thank you for the update on your view about the Russel paradox.
It show in a very clear way that mathematics is a "only"... a language and not absolute true like Plato said many years ago.
I am glad to tell you that mathematics was change already dramatically few years ago in the direction that you are working.
I am really sad and sorry that almost nobody know or talk about it.
Yours
Moshe
:frown:
Thank you moshek,
I add here some response from another forum on this subject:
Lama, the (I guess it is a) paper you linked to in your last post says absolutely nothing whatsoever about a new view. Russell's paradox is indeed a paradox, and anybody who does not accept paradoxes must consider flawed the method by which they are arrived at (and then find the flaw). Yes, the theory is flawed, if Russell can exploit it to form the paradox. No, it's not necessarily flawed, if Russell misused it to form the paradox.
Your "new point of view" is not a point of view; it is a rejection of another point of view, ostensibly because Russel exploited it to form the paradox. But the flaw you identify is not a flaw, and it is not the flaw that Russel actually exploited.
Dear y_feldblum,
Russell's paradox cannot be defined in an excluded-middle reasoning, because it dies before it is even borne.
In short, an element, which has no unique and well-defined self-identity, cannot be used to produce any paradox.
If you do not agree to what I wrote above, you have to demonstrate in a detailed way, how an element that has no self-identity can be used to produce a paradox in an excluded-middle logical reasoning system.
If you cannot do that, then you have no logical argument to be based on.
Also please be aware to the fact that the set that includes all (by using the word all we get a self reference of something to itself) of the elements that have no well-defined and unique identity, is nothing but a false statement in an excluded-middle logical reasoning, exactly as the statement
a = not_a is nothing but a false statement in an excluded-middle logical reasoning system.
In short, no false result can be used (or being exploit in your words) as a logical basis to produce a paradox in an excluded-middle logical reasoning system, or in other worlds: no false statment can be considered as a paradox in an excluded-middle logical reasoning system.
a is a if and only if it is not_a is not a paradox but a false statement, exactly as a is not_a is nothing but a false statement.
In other words: (a is a if and only if it is not_a) is (a is not_a).
Therefore Russell’s paradox cannot be defined in an excluded-middle logical reasoning, and this point of view is defiantly a new point of view on what is called “Russell’s Paradox”.
Any comments to post #115?
On Russell’s First Paradox and The
Excluded-Middle Logical Reasoning
Doron Shadmi
Abstract
For more than 100 years the first paradox of Russell is considered as a problem in the foundations of what is called Naïve Set-Theory.
In this short paper we show that this paradox is based on elements that have no unique self-identity, and we can conclude that Russell's paradox cannot be more then a false statement in the framework of excluded-middle logical reasoning.
We also show that excluded-middle logical reasoning framework is a limited logical system.
Keywords: Unique self identity, Excluded-middle logical reasoning,
False statement, Limited logical system.
Russell's first paradox by standard logical reasoning:
( http://www.wikipedia.org/wiki/Russell%27s_paradox )
Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A. In the sense of Cantor, M is a well-defined set. Does it contain itself? If we assume that it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, than it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to a contradiction. So this must be a contradiction in the underlying theory.
A new point of view on Russell's first paradox:
In excluded-middle reasoning, each element must have a one and only one unique identity.
An element without a unique identity cannot be a participator in the excluded-middle "game".
Russell's paradox arises because we let to an element, which has no unique identity, to be a participator in our "game".
For example:
The identity of the barber of Seville cannot be defined because it is based on self contradiction which is:
1) He is from Seville.
2) He is a man.
3) He shaves all of the men in Seville (which means: he is included)
4) Only if they do not shave themselves.
By this last condition he contradicts its own identity because:
To shave all (which means: he is included) of the men in Seville only if they do not shave themselves, means that all is not_all (or a = not_a ).
The same contradiction of self identity, can be shown in the set that includes all of the sets only if they do not include themselves as their own members.
To include all (which means: it is included) of the sets only if they do not include themselves, means that all is not_all (or a = not_a ).
An element which has no self and unique identity cannot be a legitimate participator in an excluded-middle logical reasoning.
Also please be aware to the fact that the set that includes all of the elements that do not have well-defined and unique identity, has a unique self identity, and we can conclude that no one of the existing members of this set can be a legitimate participator in an excluded-middle logical reasoning system (we also can conclude that these existing members are beyond the domain of an excluded-middle logical reasoning system, which means that excluded-middle logical reasoning system is a limited logical system).
In short, no false result can be used as a logical basis to produce a paradox in an excluded-middle logical reasoning system, or in other worlds: no false statement can be considered as a paradox in an excluded-middle logical reasoning system.
a is a if and only if it is not_a (it means that a contradicts its own self identity) is not a paradox but a false statement, exactly as a is not_a is nothing but a false statement.
In other words: (a is a if and only if it is not_a) is (a is not_a).
M is M if and only if it is not_M is nothing but a false statement.
Therefore Russell's paradox is not defined within excluded-middle reasoning.
Any comments to post #117?
pallidin
Jul4-04, 05:54 PM
(( Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A.))
This is not possible, as A becomes automatically void in any other set by virtue of being non-existent in it's own set.
If you are part of the Army, and I kill you, are you still a part of the Army?
In an excluded-middle reasoning an element (set, number, ...) can have simultaneously a one and only one unique name (identity).
And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.
The identity of an element is its literal name like: a number named 'pi', a number named 'e', a number named '1', a number named '0', a set named 'all_sets_that_do_not_contain_themselves' ... and so on.
Strictly speaking, a well-defined element in an excluded-middle logical reasoning system, cannot be but an element that has a one and only one unique literal name.
Now, the set that includes 'all of the elements that do not have well-defined and unique identity', has a unique self identity.
Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well defined set are false , neither true nor false, contingently true or false etc.)
There is here a positive approach of Godel's incompleteness theorem, which says: Within any consistent system, there can be found at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.
In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:
Each consistent system includes within it the seeds of its paradigm shift, and in my opinion this is the essence of the Language of Mathematics.
matt grime
Jul5-04, 07:06 AM
If Russell's paradox cannot be defined, how are you able to reason about it and, erm, define it? Your position is itself paradoxical.
You are not adding anything new to the arguments of this paradox, just inventing new terms and seemingly misapplying existing ones, however the poor presentation means it is hard to decide what you are trying to say some times.
Why do you insist on saying 'excluded middle logic' is limited as if this is somehow disturbing news? Every mathematical system is limited by and to its axioms. There are systems where excluded middle isn't used (see Martin Hyland's examples of topoi). Note, we are not giving credence to any of your attempts ot defining logical systems since you have not produced anything that we can consider to be consistent, or readable.
Hi Matt,
Long time no see.
If Russell's paradox cannot be defined, how are you able to reason about it
In my paper I show that what is considered as a paradox in naive set-theory, is no more then a false statment.
Please show me where can I find similer interpretation to this "paradox".
But the main point of my paper is about Godel's incompleteness theorem.
You can find it at post #120.
Alkatran
Jul5-04, 09:01 AM
I don't know if this was covered yet, but you guys seemed to be confused about what "not"ing a number does in programming.
not X = -X -1
The value of true is -1 and false is 0:
not false = true = -0-1 = -1
not true = false = --1 -1 = 0
The operation that would switch all the bits of a number is XOR.
X XOR True = (X with all 0s switched to 1s and 1s switch to 0s)
X XOR False = X
For example:
If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all
If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all
Some conclusions:
a) The self identity of the Barber of Seville is based on the false statement all = not_all.
b) Self identity, which is based on a false statement, is no more then a false statement.
c) No false statement is a paradox in excluded-middle reasoning.
d) Therefore Russell's paradox is not defined in excluded-middle reasoning.
In general we can conclude the above about any self-referenced definition, which includes in it all condition.
If an all condition is omitted form a self-referenced definition, then the possibility of self identity as a false statement, is avoided in an excluded-middle reasoning.
If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?
In logic we can say that our true result is a false statemant.
This is the reason why some false reuslt can be found in our logical system.
Only the true stands behind any result.
Also in excluded-middle reasoning any examined concept cannot have more than one unique identity,
so a = not_a cannot be but a false statemant (which is the true reuslt) in this case.
CrankFan
Jul10-04, 02:54 AM
"A new point of view on Russell's first paradox..." -Lama
Maybe you missed the part in the text you linked which said that:
"In Cantor's system, M is a well-defined set."
And that's pretty much all you need to know, that given the same assumptions that Cantor was making (in particular the abstraction principle) the Russell Set is perfectly valid. That you might want to add new rules, which further instruct what is and what isn't a set in naive set theory is immaterial, because once you've added those additional rules you would no longer be talking about naive set theory.
http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Naive_set_theory#Specifying_sets
When you say things like:
"In excluded-middle reasoning, each element must have a one and only one unique identity. An element without a unique identity cannot be a participator in the excluded-middle "game". -Lama
Not only is mumbo-jumbo like this nearly incomprehensible, not only is it not anything close to a proof that the Russell set isn't well defined in terms of naive set theory, but it's also completely irrelevant because phrases like "a participator in the excluded middle game" and "unique self identity" aren't known well enough for you to use them before defining precisely what they're supposed to mean in terms that everyone can understand.
I can guess your intended meaning, but if that's what you expect us to do, then you shouldn't believe that what you've provided is rigorous or a proof of anything. It's more like a guessing game.
Naive set theory is not foundational in any other sense than historically. Modern set theories don't have classical antinomies, like the Russell set. However, most introductory set theory texts will discuss them for the sake of describing key historical developments and motivating discussion about how we can properly axiomatize a set theory so that it retains the richness of naive set theory and also avoids classical problems.
Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.
http://en.wikipedia.org/wiki/Russell's_paradox#Set-theoretic_responses_to_the_Russell_Paradox
[edited for typos]
No proposition can make a statement about itself...
If we look at this propositoin, we can say that within an excluded-middle reasoning, if a self reference of a proposition changes the propositon, then and only then it cannot be refered to itsef, because in an excluded-middle reasoning, each element has exactly one and only one uniqe identity.
By tautology x = x means: x is itself, otherwise we cannot talk about x.
Now we can ask if a teotology is also recursive, for example: x = x = x = ...
If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.
So, Russel's paradox is like if by teotology we examine if x is not_x or x = not_x , which is no more then a false statement from an exluded-middle point of view.
In an excluded-middle reasoning no false statement is a paradox.
Again:
The element x_AND_not_x cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.
Therefore Russell's Antinomy is nothing but a false statemant and not a paradox in excluded-middle framework.
note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic
'=' is used here for the tautology of a = a.
a = not_a is no more than a false statment in excluded-middle reasoning.
A and not_A
As usual, you miss the point.
A and not_A cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.
Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.
Our true result in this case is no more then a false statement, and all the big affords that professional mathematicians like you put in their theories to avoid this "paradox", are no more than a full gas in neutral.
arildno
Jul10-04, 05:19 AM
Lama:
Are you willing to let yourself be tested by members on this forum in order to establish whether or not you have understood anything in standard maths?
So far, you have given no indication that you possess any such understanding.
Your own ideas would be looked at more closely if it could be established beyond any doubt that you understand what standard math is.
Hi arildno:
If you do not want to understand that x_AND_not_x is beyond (cannot be well-defined, and it means that the "paradox" cannot be defined) the domain of x_XOR_not_x (which is the basis of an excluded-middle reasoning), then your basic attitude, in my opinion, is to be no more then a full time job bodyguard of The language of Math, and (as I see it) you do not give yourself any chance to see fundamental things from a different point of view.
Maybe you missed the part in the text you linked which said that:
"In Cantor's system, M is a well-defined set."
So what if Cantor thought that M is well-defined in his system.
The Language of Mathematics is not based on gurus, but on fundamental concepts that are never beyond re-examination.
I look at this "paradox" from an included-middle reasoning, and the affect is similar as if I look on 2-D system from n>2-D system.
Someone who looks on some system from a first-order higher level of reasoning system (where 2-D reasoning system is only a proper sub-system of it, and I clearly show it in my papers) can easily show new interpretations to fundamental concepts of the Language of Mathematics.
If you stick to the standard 2-D reasoning, you will never understand my work.
I made my move to new points of view that re-examine the most fundamental concepts of this beautiful language.
Take for example persons like Matt Grime, which in my opinion make here a very good job as the bodyguard of Math.
It took me some time (almost 2 years) to understand that I am talking to a full time job bodyguard, so now I take what I take and I do not care anymore that full time job bodyguards do not want to or can’t understand my work.
You, Matt Grime, CrankFan, Master Coda, Hurkyl, kaiser soze, Ahrkron , and more full time job bodyguards of Math, did not show even a little step to see things from new points of view on the most fundamental concepts of the language of Math.
Form my side, I clearly an simply show why Standard-Math approach does not hold in these most fundamental concept.
I learn my mistakes, and I am trying to improve the basis of the reasoning of my work, but because all you do is to be full time job bodyguards, you do not distinguish that.
For example, please look at the attitude of Ahrkron to my work:
http://www.physicsforums.com/showpost.php?p=243538&postcount=37
In my opinion this is nothing but a poor, non-detailed and limited approach to someone's ideas.
More examples:
Please look at the attitude of Matt Grime to my work:
I wrote to him:
http://www.physicsforums.com/showpost.php?p=242567&postcount=28
As an answer I got:
http://www.physicsforums.com/showpost.php?p=242639&postcount=29
Another example of Matt Grime's attitude:
http://www.physicsforums.com/showpost.php?p=250748&postcount=41
And my non-friendly reply to him:
http://www.physicsforums.com/showpost.php?p=250819&postcount=42
that he chose to ignore.
Also be aware to the name 'CrankFan' that can say a lot about his limited attitude to the possibility if new interpretations of fundamental concepts of the Language of Math.
All I asked in 'Theory development forum' is a little more flexible approach that can examine "well-defined" terms from (time to time) a new point of view.
What I have found is a community of hard minds that do not want anyone to change the fundamentals of their religion.
In short, I do not accept Cantor's M definition, and I clearly and rigorously show why I do not accept it.
No full time job bodyguard can understand it.
geistkiesel
Jul10-04, 11:40 AM
"A new point of view on Russell's first paradox..." -Lama
Maybe you missed the part in the text you linked which said that:
"In Cantor's system, M is a well-defined set."
And that's pretty much all you need to know, that given the same assumptions that Cantor was making (in particular the abstraction principle) the Russell Set is perfectly valid. That you might want to add new rules, which further instruct what is and what isn't a set in naive set theory is immaterial, because once you've added those additional rules you would no longer be talking about naive set theory.
http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Naive_set_theory#Specifying_sets
When you say things like:
"In excluded-middle reasoning, each element must have a one and only one unique identity. An element without a unique identity cannot be a participator in the excluded-middle "game". -Lama
Not only is mumbo-jumbo like this nearly incomprehensible, not only is it not anything close to a proof that the Russell set isn't well defined in terms of naive set theory, but it's also completely irrelevant because phrases like "a participator in the excluded middle game" and "unique self identity" aren't known well enough for you to use them before defining precisely what they're supposed to mean in terms that everyone can understand.
I can guess your intended meaning, but if that's what you expect us to do, then you shouldn't believe that what you've provided is rigorous or a proof of anything. It's more like a guessing game.
Naive set theory is not foundational in any other sense than historically. Modern set theories don't have classical antinomies, like the Russell set. However, most introductory set theory texts will discuss them for the sake of describing key historical developments and motivating discussion about how we can properly axiomatize a set theory so that it retains the richness of naive set theory and also avoids classical problems.
Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.
http://en.wikipedia.org/wiki/Russell's_paradox#Set-theoretic_responses_to_the_Russell_Paradox
[edited for typos]
i avoied this thread for a lot of reasons, but when i styarted slowly to difest the mater what Lama has to say, that was once gobble-gobble as was everybody elses's post, I started to get the drift. It isn't listening to th Barber if seville, but it also isn't what you described the matter to be, at least not to me and I am a slow learner.
quote Lama:
If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all
If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all
This isn't difficult to understand, and it isn't gobbledy gobble. Nullify a unique aspect of something and it isn't unique any more. Adios paradox, right? I had only heard of the paradox peripherally, and could barely grasp the ssence of understanding the "paradox" and now I understand the negated paradox. A learned piece of history: Russel wasn't "everything in logic and reason" was he? And another really nice thing about it is that 2 + 2 = 4, most of the time.
This isn't difficult to understand, and it isn't gobbledy gobble
Thank you dear geistkiesel for allowed yourself to be opened to another point of view of this "paradox".
I can understand why professional mathematicians do not want to look on fundamental mathematical concepts from a different point of view.
In My opinion, the most basic important thing of any formal or informal language, is the ability of an open dialog on any concept of any formal or informal language.
If this basic and important thing is omitted from some language, then (in my opinion) this language is a dieing language.
matt grime
Jul12-04, 05:00 AM
No, Doron, we don't want to look at basic objects from different points of view, which is why there is only one set theory... oh no, look at that there isn't just one set theory, there are many, damn, wrong again. If you don't believe me try looking up some stuff in journals of computational mathematics. The last one I looked through had about 5 different set theories mentioned in the first 3 papers. Once more your lack of knowledge and unwilllingness to open yourself to others' ideas has led you to believe in something false. There are also many difference logic types too, not that you seem to accept this.
Would you care to offer any mathematical arguments or examples to show where your opinion is true? And where the refusal to accept it has led to problems?
contiune...
On Godel's incompleteness theorem:
In an excluded-middle reasoning an element (set, number, ...) can have simultaneously a one and only one unique name (identity).
And we do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.
The identity of an element is its literal name like: a number named 'pi', a number named 'e', a number named '1', a number named '0', a set named 'not_all_sets_that_do_not_contain_themselves' ... and so on.
Strictly speaking, a well-defined element in an excluded-middle logical reasoning system, cannot be but an element that has a one and only one unique literal name.
The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.
Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)
This is a positive approach of Godel's incompleteness theorem, which says:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.
In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:
Each consistent system includes within it the seeds of its paradigm shift, and in my opinion, this is the essence of the Language of Mathematics.
Hi Matt,
Once more your lack of knowledge and unwilllingness to open yourself to others' ideas has led you to believe in something false.
Do you understand my ideas? Prove it!
Would you care to offer any mathematical arguments or examples to show where your opinion is true? And where the refusal to accept it has led to problems?
The problem is never in some system and always in the limited minds of the person how refuses to look at fundamental things from different points of view.
Since you never showed any motivation to understand my work about fundamental concepts of the language of Mathematics, you cannot understand, for example this paper: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
which answers to your last question.
The last one I looked through had about 5 different set theories mentioned in the first 3 papers
Another typical response of you.
Instead of compare between the contents of these theories, to my theory, you count how many of them can be found, which is a very deep approach.
Also read: http://www.physicsforums.com/showpost.php?p=253586&postcount=129
matt grime
Jul12-04, 08:33 AM
Ahem, the reason I, and no one else it appears, can understand your "theories" is because they do not make any sense for many reasons. How can anyone demonstrate understanding of that which cannot be understood. You don't even appear to be able to understand the fact that your allegations about mathematics are without foundation, so why should we listen to the arguments based upon faluty premises?
Ahem, the reason I, and no one else it appears, can understand your "theories" is because they do not make any sense for many reasons. How can anyone demonstrate understanding of that which cannot be understood. You don't even appear to be able to understand the fact that your allegations about mathematics are without foundation, so why should we listen to the arguments based upon faluty premises?
This superficial and non-detailed monolog was infront Matt Grime's mirror.
And because he thinks that to talk to a mirror is a dialog, he cannot understand what is a dialog.
An example of a dialog between Muddler an Lama can be shown here:
http://www.physicsforums.com/showthread.php?t=33687
I think that you have something smurt to say about http://www.physicsforums.com/showpost.php?p=254672&postcount=133
Do you now why i am so nice to you?
because in this year you are going to get your phd in Mathematics, and then you are going to be a teacher for a new generation of mathematicians that most of them will not survive your limited doctrine and become your duplicates.
If there are too many mathematician that have a limited doctrine like you, then (in my opinion) there is a real danger that in couple of generations this beautiful language will die because of luck of creative and opened minds, which understand that no language (formal or informal) can survive without an opened dialog, that allowed itself to re-examine any of its fundamental concepts.
No full time job bodyguard of the Language of Mathamatics, can understand it, and in my opinion, you are one of the best.
Chronos
Jul12-04, 02:32 PM
I'm a little fuzzy about the logic here. Sounds a lot like Eubulides 'Liars Paradox', which has been around for about 2400 years. For amusement see
http://www.utm.edu/research/iep/p/par-liar.htm
Hi Chronos,
Yes Russell's first paradox is based on the principle of the liar's paradox.
For example:
X = Liar
Y = Honest
X cannot have any property of Y, and Y cannot have any property of X, In an excluded-middle reasoning or in other words:
A true Liar cannot say directly (by using the word "I") or indirectly (by using the word "ALL") that he is a Liar, because he can’t say the truth.
A true Honest cannot say directly (by using the word "I") or indirectly (by using the word "ALL") that he is a Liar, because he can’t lie.
X has a one and only one unique self-identity.
Y has a one and only one unique self-identity.
The logical condition between X and Y in an excluded-middle reasoning is : X_XOR_Y.
The paradox is based on X_AND_Y but because any well-defined element in an excluded-middle reasoning cannot have more than one unique self-identity, then no element, which its identity is based on X_AND_Y is well-defined in the domain of excluded-middle reasoning.
It means that the Liar's paradox (and also Russell's first paradox) is not well-defined concept within excluded-middle reasoning.
The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.
Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)
This is a positive approach of Godel's incompleteness theorem, which says:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.
In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:
Each consistent system includes within it the seeds of its paradigm shift, and in my opinion, this is the essence of the Language of Mathematics.
matt grime
Jul13-04, 06:02 AM
But, Doron, you don't want to learn about other views of mathematics, as you've made abundantly clear. Your idea of a dialogue is expounding your views, not in learning about mathematics. As you are the one who wants the dialogue, you'd think you'd at least be prepared to listen to others?
Matt,
I am listening to you and I write my non-standard ideas, which sometimes refer directly to what you say, sometimes refer indirectly, and sometimes take me to a new place where I can find ideas which are beyond the scope of the original starting point.
Then I write my non-standard Ideas in non-standard methods, because the standard methods most of the time cannot express correctly my non-standard ideas.
You have in your mind the standard concept of how the Language of Mathematics should be expressed and developed, but I have both new fundamental ways of how this Language can be developed and new fundamental points of view on the most fundamental concepts of this language.
In this most fundamental level when the subject is a paradigm shift of some concept, and someone inviting you to a dialog in these conditions, you have to take your first steps when you are armless.
No full time job bodyguard of Math that using his standard point of view as a weapon, and come to save the holy Math from crackpots like me, will be able to understand what I have to say. :wink:
kaiser soze
Jul13-04, 03:05 PM
Lama,
You overlook several important aspects of maths. Maths is not just a language, it is also a way of thinking. Thinking mathematically is an aquired skill - no one is born with it, it is developed and refined through doing maths (and in this case maths is what mathematicians are doing!). Another important aspect of maths is abstraction - it is in fact one of the cornerstones of maths. Abstract thinking is not trivial and not "natural" but is essential to understanding mathematic concepts, constructs and realms.
I have noticed that you avoid abstraction, and try to address things graphically - while this may help in some cases, it is inappropriate in others, and deminishes understanding and grasping of abstract notions. Most of the issues you are referring to are very abstract by definition - I think that when you master (traditional) mathematical thought and abstraction your views on your current ideas will change.
Kaiser.
Dear kaiser soze,
First, thank you for your kind post.
I agree with you about what can maybe called "the art of abstraction".
My gateway to this art is triggered by visual imagination, which means that the graphic forms are only tools exactly as linear lines of symbols are tools.
When you understand something deeply, it is always in the abstract level, where tools like pictures, symbols, sounds, and so on, do not exist anymore.
In short, each person can use his favorite trigger to develop his abstraction skills, and no method is better then the other.
The problem arises when some community of people does not distinguish between its tools and the internal cognitive abstract states of mind.
Maths is not just a language
A Language is many things in many levels of abstraction, which can be expressed and developed by dance, picture, writing, smell, music, movement, touch, sound, silence, colors, taste, light, sport ... and infinitely many other ways (and combinations of them).
And no community of people can put this natural power in boxes and tell us what is the right way of thinking and what are the right tools that we have to use, if we want to develop abstract cognitive skills.
Only one thing has to be developed: the art of internal/external dialog, and the rest is based on it.
terrabyte
Jul13-04, 04:41 PM
Each consistent system includes within it the seeds of its paradigm shift, and in my opinion, this is the essence of the Language of Mathematics.
mostly this is only so because we choose to define it that way. For every created system we have elements that we define having values. this is all well and good until we come to the point where we start comparing and contrasting these values against each other. suddently we have to define not only what elements "Are" but also what they "Are not". in doing so we create these "seeds" for paradigm shift that you speak of. they're not necessarily inherant in the system, they're just a result from applying the system to define something.
indeed you'd have a hard time labelling "an honest person" if you first did not define "a lie". you could say "this person always tells the truth" but then the logical path that comes to your mind is "is there anything but 'the truth'?". there has to be or everyone would be an "honest person" and there would be no reason to define that term at all.
Hi terrabyte,
they're not necessarily inherant in the system,
I agree with you, because seeds do not strat to grew without an external trigger, like water for example.
So a paradigm shift needs both "seeds", and us as "water".
Chronos
Jul14-04, 12:43 AM
Language is inefficient. It naturally tends to be self-contained and paradoxical. Math is better, but, not entirely. It too contains self contradictions. What Godel said, in essence, is that no theory can prove 'a priori' assumptions. In effect, Godel says his theory of incompleteness, is incomplete. Ironic. Logic and math are first cousins.
By Godel we can understand that any consistent(and therefore limited) system cannot be complete(and therefore without limits), and any complete system cannot be consistent.
In short, the concepts consistent and complete are preventing/defining each other.
For example, please see this picture: http://www.geocities.com/complementarytheory/comp.jpg
As you see the two black profiles and the white vase are clearly preventing/defining each other.
Please also see http://www.geocities.com/complementarytheory/CompLogic.pdf , which is a short paper of mine on included-middle reasoning.
This is, by the way, the deep reason why universal quantification cannot be related to a collection of infinitely many things.
It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things.
CrankFan
Jul14-04, 07:13 AM
"This is, by the way, the deep reason why universal quantification cannot be related to a collection of infinitely many things." -lama
Really? Why is that? specifically?
"It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things." -lama
Ok, that's your opinion but you've not provided any real explanation (just some hand-waving about Goedel incompleteness) why we can't use universal quantification to say meaningful things about (infinite) sets, like for example the set of all positive integers:
Ax(not(x = 0) -> (not Ay (not (x = Sy))))
Any, non-zero natural number x is the successor of some natural number y.
These sentences, both the informal and almost-completely-formal version seem perfectly meaningful to me.
Previously you speculated that I was a professional mathematician. I'm not. In fact I'm not even remotely close to being a professional mathematician. I'm a guy who likes to learn mathematics when I have some free time. So I guess I'm an enthusiast, or something like that. Anyway, getting back to the topic... People aren't correcting you because they're bodyguards of math, whatever that is supposed to mean. People are correcting you because you make statements about mathematics which are either known to be false or are unsubstantiated.
matt grime
Jul14-04, 11:01 AM
"By Godel we can understand that any consistent(and therefore limited) system cannot be complete(and therefore without limits), and any complete system cannot be consistent."
that is not an accurate or succinct interpretation of what Goedel's incompleteness theorem actually states, and you have its negation correspondingly incorrect. You may wish to look it up.
matt grime
Jul14-04, 11:06 AM
"It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things."
you may wish to explain to the uninitiated what your definition of "all" is mathematically, since it does not accord with its usage in predicate calculus.
Matt and Carnkfan,
Thank you for your posts.
Please this time read all of http://www.geocities.com/complementarytheory/ed.pdf
In this short paper I clearly show another possible and non-standard point of view, which explains why, for example, a universal quantification cannot be related to R collection.
Chronos
Jul15-04, 12:08 AM
Mathematically, the term 'all' only has meaning in set theory.
Chronos
Jul15-04, 12:41 AM
Lama, one thing I am curious about. I am a bit naive when it comes to logical operators and you seem quite comfortable with them. To me, the mathematical equivalent of the 'Liar Paradox' [this sentence is false], is 'x is not equal to x'. The math that follows completely breaks down in my mind. 1 multiplied by 'x' and 2 multipied by 'x' can easily be reduced to the expression '1 = 2'.
Matt,
that is not an accurate or succinct interpretation of what Goedel's incompleteness theorem actually states
Ok, let us put it this way:
By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proved by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Now, a complete system that suppose to prove everything is inconsistent by definition, therefore no consistent system (and therefore incomplete) can prove anything, and because of this we can find within any consistent system statements that are well-defined within the framework of the consistent system, but they cannot be proved within this framework, unless we add more axioms to the system, and so on, and so on.
So if by using the word 'complete' we mean 'inconsistent', then no 'complete' system is consistent.
Let us return to the universal quantification 'all'.
As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements ,even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements, is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*[1/X] = 1
In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.
'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.
Hi Chronos
1 multiplied by 'x' and 2 multipied by 'x' can easily be reduced to the expression '1 = 2'.
Sorry, please explain this.
Russell E. Rierson
Jul15-04, 01:43 AM
Sorry, please explain this.
:wink: :wink: :wink:
A = B
A^2 = A*B
A^2 - B^2 = A*B - B^2
A^2 - B^2 = B*[A - B]
[A + B]*[A - B] = B*[A - B]
A + B = B
and
A = B
2*B = B
2 = 1
If one forgets that A - B = 0, because we cannot divide by 0. Unless you have an amazing discovery ...Lama?
A/0 is undefined.
0/0 is indeterminate.
2*X cannot equal 1*X
The interval from zero to one has an infinite number of fractions and it is a finite unit.
A = not-A ?
LIM X---> 0, X*[1/X] = 1
Some may argue with the epsilon-delta formalism of Carl Wierstrass but there is also Abraham Robinson's non-standard analysis, which puts infinitesimals on a rigorous footing.
matt grime
Jul15-04, 06:34 AM
Your second description of godel's theorem is even less succinct and still inaccurate. Perhaps you'd like to see what i can recall of it?
If S is any (set) theory in which there is a model of the natural numbers (you've never managed to include that), then there exist a proposition, P, such that both P and not P are consistent with S.
Your definition of 'all' is garbage.
Your definition of 'all' is garbage.
I told it to ahrkron and also I tell you, my work is like a mirror at the first stage.
If you understand it you can see beyond your own reflection, but in your case you see nothing but your face in it.
If S is any (set) theory in which there is a model of the natural numbers (you've never managed to include that), then there exist a proposition, P, such that both P and not P are consistent with S.
Since you cannot go beyond the standard formal definitions, you cannot understand that I am talking about x_AND_not_x which are members of set S of 'any undefined element of some (set) theory'
where S is well defined in this (set) theory.
What we get in this case is a well-defined set S as a 'Trojan horse' which includes in it elements that cannot be defined in the framework of the examined theory.
So as you see i do not stay in the original formal definition of Godel's theorem, but I use the deep principle of it to develop another point of view, which is much more interesting (in my opinion) then Godel's theorem.
In short, Godel's theorem is often used to show the limitations of a system, but my point of view is to show that any limited system actually lead us to search beyond its domain, which is a positive approach of the same idea.
Since you do not aware to the power of the philosophical thinking as a profound tool for deep mathematical fundamental ideas, you cannot see but the reflection of your formal face in my philosophical mirror.
Matt, if you think that a good mathematician is a walking encyclopedia of formal mathematical knowledge, which is used to produce results within a particular brunch of the standard framework, and he never think beyond his own domain, then your way is not my way.
------------------------------------------------------------------------------------------------
Your definition of 'all' is garbage.
Theorem: Matt's response is a garbage.
Proof:
By inconsistent system we can "prove" what ever we want with no limitations, but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proved by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Let us re-examine the universal quantification 'all'.
As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements ,even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements, is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*[1/X] = 1
In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.
'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.
If my definition of 'all' is garbage, then 0*(1/0)=1
Since 0*(1/0) not= 1, Matt's response is a garbage. QED.
Chronos
Jul16-04, 03:29 AM
hi lama.
i think i understand your point. but, i dont think you can logically prove mathematics anymore than you can mathematically prove logic. i prefer math to explain science and logic to explain philosophy. The essence of Godel was to say that all systems are self-referential [they necessarily include 'a priori' assumptions] and assumptions cannot be proven or disproven within any such system. Relativity, in this sense, may be the most profound statement about reality we are capable of grasping.
Russell E. Rierson
Jul16-04, 03:33 AM
Back to the "Fallacy of Composition" for the educational benefit of Hurkyl and co.
The properties of the whole distribute over the individual parts OF the ...whole :eek:
[WHOLE]--->[PARTS]
If W then P
W
therefore P
Properties of the parts don't necessarily become properties of the whole.
If W then P
P
therefore W
is false logic, also known as the fallacy of composition via modus ponens error. :devil: :devil: :devil:
QED :wink:
matt grime
Jul16-04, 04:27 AM
"So as you see i do not stay in the original formal definition of Godel's theorem, but I use the deep principle of it to develop another point of view, which is much more interesting (in my opinion) then Godel's theorem."
but have you proved it is true?
in any case you are demonstrating explicitly again that you are using a different definition and usage from the rest of the world in your use of "all" amongst other things, so don't correct us or tell us mathematics is wrong when an "all" is used (ie your problems with Cantor's Theorem) when you are refusing to use the words correctly, as they were intended.
tell you what: 2 > x for all x in (0,1), wow, i used the word all for a collection of infinitely many things...
you're also scoring high on the crackpot index again with your combined insistence that what you have is amazing and so much better than crappy mathematics as practised by religious dogmatic fanatic and admitting that your theory is allowed not to be very good because it's new and you aren't a raligiously fanatical adherent of limited mathematics.
in any case you are demonstrating explicitly again that you are using a different definition and usage from the rest of the world in your use of "all" amongst other things
So, do you support the idea that Mathematics is no more then a rigorous agreement between people?
"Math, in my opinion, is first of all a rigorous agreement that based on language.
Symmetry is maybe the best tool that can be used to measure simplicity, where simplicity
is the best platform for stable agreement.
Any agreement must be aware to the fact that no model of simplicity is simplicity itself.
This awareness to the difference between x-model and x-itself is the first condition for
any stable agreement, because it gives it the ability to be changed."
http://www.geocities.com/complementarytheory/CATpage.html
but have you proved it is true?
It is trivially understood that in any consistent (set) theoretical system, there is at least one well-defined set that includes any of the undefined elements of this consistent (set) theoretical system.
Actually it is an axiom, which may be called "The axiom of the paradigm-shift".
crackpot
"crackpot" is a reflexive response of a 'limited system' when it is forced to look beyond its limited domain.
In that case I am proud to be called a "crackpot".
matt grime
Jul16-04, 11:18 AM
So it's an axiom in your system?
Do you recall that you once claimed that you'd successfully answered the Collatz conjecture because you'd "proven" that it was equivalent to the axiom of infinity, and was thus not provable? Have you changed you mind about axioms being true then?
Do you recall that you once claimed that you'd successfully answered the Collatz conjecture because you'd "proven" that it was equivalent to the axiom of infinity, and was thus not provable? Have you changed you mind about axioms being true then?
No Axiom can be proven within its own framework, because it is an arbitrary true, that is based on our intuitions, therefore a statement which is equivalent to an axiom, cannot be proven within its own framework, as I show in http://www.geocities.com/complementarytheory/3n1proof.pdf
Only statements that are not equivalent to an axiom can be proven within their framework.
In short, only statements that are based on axiom(s) but do not equivalent to any one of them, can be proven within their framework.
For example:
The Natural numbers are axiom's products and not the axioms themselves, so we can prove things which are based on N members' properties only if we do not need to go to the level of the axioms themselves.
We can use here an analogy, which is based on chemistry:
In order to prove something in the level of the muscles tissue, we do not need quarks level.
Chronos
Jul17-04, 05:08 AM
So it's an axiom in your system?
Even a domesticated bunny with pink eyes could see that trap... hilarious matt. But forgive lama for being logically inconsistent, he means well.
But forgive lama for being logically inconsistent
It depends on the interpretation of "what is a proof?"
My interpretation excludes the axioms of some given consistent system, because they are arbitrary true within the framework of their own system.
Their 'true' validity can be examined and proven only within some Meta framework, where they cannot be considered as axioms anymore.
In that sense, any consistent system has two sides:
a) The internal side is the consistent and therefore limited system, which is based on its own arbitrary true (self evidenced) axioms.
b) The external side which is not limited but then not necessarily consistent with this system.
c) In that sense, no axiom can be proven within its own framework, and on the same time, this axiom must be proven within some Meta-framework.
So, as we can see, an axiom is an element that always exists between internal and external frameworks.
kaiser soze
Jul17-04, 11:24 AM
Occam's Razor:
"one should not increase, beyond what is necessary, the number of entities required to explain anything"
http://pespmc1.vub.ac.be/OCCAMRAZ.html
Hi kaiser soze,
So can we ignore the butterfly wings when we construct the model of a hurricane? (http://mathworld.wolfram.com/ButterflyEffect.html)
arildno
Jul17-04, 04:27 PM
Hi kaiser soze,
So can we ignore the butterfly wings when we construct the model of a hurricane? (http://mathworld.wolfram.com/ButterflyEffect.html)
Certainly, because that effect would be included in the initial conditions, not the system of differential equations which is THE MODEL
Certainly, because that effect would be included in the initial conditions, not the system of differential equations which is THE MODEL
Differential equations are nothing but a tool here.
The Model in this case cannot be less then a combination of initial conditions AND the system of differential equations, otherwise our nonlinear phenomena cannot be understood.
In short, we can never be 100% sure what is necessary and what can be omitted, when we deal with nonlinear complexity, for example.
arildno
Jul17-04, 07:29 PM
Blather Again, Lama!
kaiser soze
Jul18-04, 12:57 AM
Lama,
You missed the point. A common interpretation for Occam's Razor is: In science, the simplest theory that fits the facts of a problem is the one that should be selected.
Occam's Razor is a rule of the thumb for selecting theories to explain phenomena.
Kaiser.
Blather Again, Lama!
This is a very poor response, I think that you can do better than that.
You missed the point. A common interpretation for Occam's Razor is: In science, the simplest theory that fits the facts of a problem is the one that should be selected.
Occam's Razor is a rule of the thumb for selecting theories to explain phenomena.
Do you really think that a non-trivial problem or phenomena are some clear objects that are waiting for us to explain them?
When we cut out things we change our initial conditions, and if our explored system is sensitive to initial conditions then your "simplest" theory misses the point.
In short, the quality of a theory is much more important than the quantity of its components, when we deal with non-trivial systems.
Any way, you did not explain in details, why do I have to use Occams' Razor on my arguments, so please do that, thank you.
Russell E. Rierson
Jul18-04, 03:56 AM
Q1:
If the barber shaves those, and only those men who do not shave themselves, then does the barber shave himself?
Q2:
If an assertion A, is true and its negation, ~A is also true, it becomes a form of the "liars paradox".
Suppose a person called X, stands up and says, "This assertion is false."
Let S denote the statement uttered; let p be the proposition the person makes by uttering S. Then the utterance of the phrase "This assertion" refers to the claim p. It follows that, in uttering the words "This assertion is false," X is making the claim "p is false". Thus , p and "p is false" are one and the same:
p = [p is false]
By making the claim, X is implicitly referring to the context in which the claim is stated. Let c symbolically represent the context for which the sentence refers.
X's uttering of the words "This assertion" refers to the context, c, which entails p.
[c entails p]
That is to say, p must be the same as [c entails p] due to the fact that X is referring to both p and [c entails p] via the utterance of the phrase "This assertion."
If X's assertion is true then [c entails p] is true
p = [p is false]
[c entails p is false] is true
This creates a contradiction, ergo X's claim that [p is false] is false.
[c entails p is false] is false
This appears to be the same contradictory state of affairs as in the previous cases of the Liars Paradox.
Conclusion?:
c cannot be the appropriate context.
Consequently, the paradox becomes a theorem/demonstration.
When X utters the Liar sentence, X is uttering a falsehood, and the context in which the claim of falsehood is made cannot be the same as the context in which the Liar sentence S, was uttered...
:eek: :eek: :eek:
Thus context c, becomes a subjective/qualia operator.
kaiser soze
Jul18-04, 06:47 AM
Lama,
Tell me what phenomenom you are trying to explain using your theory(ies) and I will explain how Occam's razor rule of the thumb should be applied.
Kaiser.
a) Godel's incompleteness theorem.
b) The limit concept
c) the universal quantification concept.
d) The inifinty concept.
kaiser soze
Jul18-04, 07:53 AM
Ok, these are not phenomena, and in any case they can be expressed or defined using simpler terms than your explanations; thus by Occam's razor these simpler terms and definitions should be prefered.
Kaiser.
and in any case they can be expressed or defined using simpler terms than your explanations;
Please demostrate your arguments by showing side by side my definitions and the standard definitions, that by your argument have the same interpretations of mine (to post #176 concepts) but in simpler ways.
If you cannot do that, then you demostrate that you do not know what are you talking about.
kaiser soze
Jul18-04, 09:19 AM
If you do not already know the "standard" definitions and interpretations of the issues you have stated then you are the one who does not know what he is talking about...
Kaiser.
I cannot give a new interpretation to a fundamental standard interpretation if I do not know it.
Since I give new interpretations to post #176 concepts, all of them are based on deep understanding of the standard interpretations of these concepts.
If you do not agree with me (which is perfectly ok), you have to demonstrate why in details.
kaiser soze
Jul18-04, 12:59 PM
Lama,
Tell you what, convince me that you understand fundamental mathematical definitions and I will gladly enter into discussion with you about them. Please provide a mathematical definition for issue (b) in post #176.
Kaiser.
Definition is not the point in this philosophical level, understanding is the point in this philosophical level.
Please read all of http://www.geocities.com/complementarytheory/ed.pdf
where you can find my new intetpretation to b,c,d subjects of #176.
If you do not like my answer then please define 'definition' and I'll try to give my answer by your definition to 'definition'.
a,b,c,d of #176 can be found in http://www.iidb.org/vbb/showpost.php?p=1716483&postcount=76
kaiser soze
Jul19-04, 01:14 AM
Lama,
I need to be sure you understand fundamental mathematical definitions as they are understood by mathematicians. You have stated many times before, that one should keep an open mind and develop criticism based on understanding. I am not sure you understand fundamental mathematical concepts, yet you criticise them on a daily basis.
Please provide a mathematical definition to issue (b) of post #176. At this stage your interpretations are irrelevant, I am talking about mathematical defitinions that would be accepted by any mathematician.
Kaiser.
Dear kaiser soze,
Since I am not a professional mathematician, my best definition at this stage is:
A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.
It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.
By using the word "leap" we mean that we have a phase transition from state A to state B.
There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.
A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.
From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.
If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.
Please look again on My Riemann's Ball example, in page 3 http://www.geocities.com/complementarytheory/ed.pdf
Thank you.
Lama
kaiser soze
Jul19-04, 02:20 AM
Lama,
By any standard this is not a mathematical definition of a limit. This is what I mean when I say that you lack the knowledge of fundamental mathematical definitions. A definition of a limit can be found in any calculus text book - usually this is the subject of the first lecture in undergarduate school.
Kaiser.
I think that we do not understand each other.
I gave you MY definiton of the limit concept.
Now, please give the standard definition for this concept.
After you give the standard definition, then we shall compare between
the two approaches.
Any way do you agree with http://mathworld.wolfram.com/Limit.html definition?
kaiser soze
Jul19-04, 03:08 AM
off course I agree with this definition. I meant for you to provide the defintion for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical defintions before we can move on to your definitions.
Kaiser.
Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.
Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."
Weierstrass took this informal definition and gave this rigorous arithmetical definition:
The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence
(|S-Sn| < e for any N < n).
kaiser soze
Jul19-04, 01:16 PM
Lama,
Very good! now based on the definition you provided, which is a correct mathematical definition please find out the limit of the following sequence:
0.9,0.99,0.999,0.9999,0.99999,....
Kaiser.
Dear kaiser soze,
Now please listen to what I have to say.
First please read http://www.geocities.com/complementarytheory/9999.pdf
(which is also related to your question) before we continue.
Thank you.
Lama
I disagree with the intuitions of Weierstrass, Cauchy, Dedekind, Cantor and other great mathematicians that developed the current mathematical methods, which are dealing with the Limit and the Infinity concepts.
And my reason is this:
No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.
In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf
Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.
Therefore no such constant can be considered as a limit of the above collection.
It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B.
So here is again post #184:
Since I am not a professional mathematician, my best definition at this stage is:
A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.
It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.
By using the word "leap" we mean that we have a phase transition from state A to state B.
There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.
A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.
From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.
If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.
Thank you.
Lama
'Any x’ is not ‘All x’
By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proven by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Let us examine the universal quantification 'all'.
As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*[1/X] = 1
In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.
'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.
kaiser soze
Jul20-04, 12:11 AM
If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.
In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. A sequence does not have a limit, if it is not approaching some constant, for example the sequence 1,2,3,4,... does not have a limit, it disperses to infinity.
Kaiser.
Since the context here is new mathematics I aloud myself to share with you this problem:Few month ago i found in one of the popular books about Wittgenstein a quote that Wittgenstein describe a possibility to create new mathematics with the geometry of Klein bottle, I am searching now for the exact reference if anyone can help me with that I will thank him
Moshek :surprise:
If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.
1 as the limit of the sequence 0.9,0.99,0.999,0.9999,0.99999,.... is based on an ill intuition about a collection of infinitely many elements that can be found in infinitely many different scales, as can be clearly understood by posts #190,#191,#192.
You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,.... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,.... }_XOR_{1} connection.
Maybe this example can help:
r is circle’s radius.
s' is a dummy variable (http://mathworld.wolfram.com/DummyVariable.html)
a) If r=0 then s'=|{}|=0 --> (no circle can be found) = A
b) If r>0 then s'=|{r}|=1 --> (a circle can be found) = B
The connection between A,B states cannot be but A_XOR_B
Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1.
You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 --> 1
A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.
terrabyte
Jul20-04, 03:42 AM
what kaiser is saying is that 1 is the limit, but 1 is not included in the set of .9+.09+.009...
i THINK what Lama is saying is:
intuitively the number this "approaches" is 1, getting infinitely close to but never reaching it. but actually the number it really "approaches" is .999...
in other words you're both saying the same thing
...getting infinitely close...
"getting close" is reasonable.
"getting infinitely close" is not reasonable, because nothing can be closer to something when something is some constant and the "closer" element is one of infinitely many elements that can be found in infinitely many different scales.
Moscowjade
Jul21-04, 09:41 AM
I think you are confused about how something can be quantized by NOT, when x = x is a statement of IS. The paradox lies in the perception of non-existence, which by its own definition, cannot exist.
The paradox comes from trying to divide 1 by zero, because zero goes into one zero times. All numbers can be divided by one, so it is irrelevant to suggest that all other numbers can't be divided by zero either. However, we use zero in order to quantize one because without it, you can't quantize infinity. One always equals one. So, zero or NOT is just a tool of perception we use to quantize one, or IS. IS always equals IS just as x always equals x. X does not replicate from x, but when quantized with the perception of _x, x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x=x= x ,ad infinitum. The paradox is just a perception of x not being x.
In my mind, zero can't change one, but it can bend your perception of one where
1/0 = C, where C is zero rippling the perception of one into 10101010101010101010101010101010101010101010101010 101010101010101010.
The paradox lies in the perception of non-existence, which by its own definition, cannot exist.
There is no 'own definition' in the first place; therefore there is no paradox.
If your reply is based on my first post of this thread then ignore it and instead
please read: http://www.geocities.com/complementarytheory/Russell1.pdf
Thank you,
Lama
arildno
Jul22-04, 11:29 AM
Just a question, Lama:
You haven't by any chance read Hegel's "Wissenshaft der Logik"?
Several of your ideas seem to be in tune with Hegel's ideas..
Hi arildno,
No, I did not read any of Hegel's work.
Thank you for the information, I'll try to find an English version of it.
Can you give us some example, which shows the similarity between Hegel's work and my ideas?
Thank you.
Lama
arildno
Jul22-04, 04:15 PM
Hi arildno,
No, I did not read any of Hegel's work.
Thank you for the information, I'll try to find an English version of it.
Can you give us some example, which shows the similarity between Hegel's work and my ideas?
Thank you.
Lama
Well, it's been years since I read Hegel, but he opposed, for example, to what he found was a philosophically incorrect concept of the limit.
(Basically, he meant the limit concept involved a "qualitative change" in the fundamental nature of the number)
Now, this is possibly of little interest/relevance to your own concepts, but I sensed a "resonance" of your ideas with Hegel's, rather than any specific concepts I could pinpoint at the instant.
Well my interpretation of the infinity concept is based on the complementary relations between the symmetry concept and the information concept.
Please read my answer to Gza: http://www.physicsforums.com/showpost.php?p=263204&postcount=69
and also my answer to Hyrkyl: http://www.physicsforums.com/showpost.php?p=263942&postcount=71
Also you can find my work in: http://www.geocities.com/complementarytheory/CATpage.html
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