The Foundations of a Non-Naive Mathematics

[Gokul43201]

Wow ! What fun !

 

[matt grime]

What on Earth do you mean by 'at last'? And why the abuse of type face and colour? You didn't at any point honestly think I had any interest in what someone who patently has no foraml training in, nor desire to learn about, mathematics had to say about their uneducated version of mathematics? Notice the key one there: you have only ever wanted to put forward your twisted view of things. You are not ramunajan, ok, you are a crank who does not understand the first thing about mathematics and can't seem to realize that the following statements are not consistent.

X is defined by s statement of the form P(X)
Y is defined by statements of the form P(Y), Q(Y), or Z(Y)
nothing that qualifies to be a Y can be an X.

 

[CrankFan]

When Matt said:
"I can write one down and demonstrate it's a bijection, eg N to N\{1} given by x -> x+1, so look, by example there can be."

Lama responded:
"When you are asked to explain why there is a bijection between some set to a proper subset of itself, the answer cannot be: "because set A has a bijection with its subset B, that can be written (for example) as f:A -> B"

First f:A->B doesn't imply that f is a bijection.

Second, Matt didn't merely suppose the existence of a bijection, he showed you how to construct it, f(x) = x + 1.

You ignored his perfectly reasonable demonstration and then started complaining, why?

Which confuses me. If you really don't know what a bijection is then why are you wasting your time redefining the foundations of mathematics? Instead, maybe you should bury your head in some math books so that you can learn why, for yourself.

A function that is both one-to-one and onto is a bijection. (**INFORMALLY** it establishes an exhaustive pairing of elements between the domain and range). Now, armed with this definition, you can use it to verify to your own satisfaction that the function matt described above is a bijection.

The beauty of math is that no one has to convince you of anything. Assuming that you're working with the same definitions that everyone else is (which is critical) then the conclusions follow naturally. In your case I don't think you are understanding the basic definitions. For example, it was a major mistake of yours to think that f:A->B alone, implies that f is a bijection.

Basic mistakes like this tell us that you don't understand the subject very well, so why should we take your grandiose claims seriously? when you've shown that you don't even understand the basics? After all, you want to be a guy who rewrites the foundations of mathematics. If I had such a lofty goal, at the very least I'd familiarize myself with the current foundations of mathematics, before characterizing them as "naive".

Your paper is most understandable at the beginning yet shows the same kind of basic errors discussed above as early as page 2. The paper becomes more and more incomprehensible as you develop your own poorly defined and intuitive pseudo-math vocabulary in favor of well defined and established mathematical terms.

 

[Lama]

it was a major mistake of yours to think that f:A->B alone, implies that f is a bijection.

CracnkFan, you missed the main point of my argument.

I used f:A->B here as the most general notation of Standard Math mapping, where some case of it is a bijection (1-1 and onto).

Your paper is most understandable at the beginning ...

If you do not understand my paper, then you cannot show us any meaningful conclusion about it.

Let us use my glasses analogy again:

Let us say that blue means a new thing.

You have yellow glasses; therefore you see any blue (new) thing as a green thing (which is not a new thing).

I asked you to take off your yellow glasses before you look at my blue (new) things.

You ignore my request and say: "nothing is new here, don't you see?
what you call blue is nothing but well-known green things!"

Let us say that you take off your yellow glasses and then for the first time you can see a natural (by transparent glasses) blue thing, but you cannot understand it because any blue thing can be understood by you only if it is a green thing.

I'll say it clear and load for the first and last time: We are in a Theory Development forum where every concept is not beyond re-examination.

You come to this forum wearing your yellow glasses concepts about the Langauge of Mathematics, and because of this you prevent from yourself to re-examine so-called "well-defined" things.

Please use your transparent glasses in a theory development forum, and also be aware that in this case no green things can fully help you to understand natural (by transparent glasses) blue things.

If you cannot do that, then please let us not waste our time in a non-dialog.

My main approach about what is called "The Langauge of Mathematics" cen be found in the front page of my website here: http://www.geocities.com/complementarytheory/CATpage.html

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Let us go straight to main points, where my theory is different from the Standard point of View.

1) Through My new point of view, any number is first of all an information-form which is based on at least {._.}_AND_{.}, where {._.}_AND_{.} is the minimal existence of any number which is not 0.

2) In any given quantity which is > 1, each number can be ordered by several internal symmetrical degrees that can be clearly shown here: http://us.share.geocities.com/complementarytheory/ONN.pdf .

3) {._.}_AND_{.} of set N, cannot be put in a bijection with proper subsets of themselves, because of {._.} that exists as an internal building-block of each N member, and I clearly show it in page 3 of http://www.geocities.com/complementarytheory/No-Naive-Math.pdf.

4) Standard Math ignores {._.} because through its point of view, any given {._.} can be defined by {.}.

5) Through my point of view {._.} cannot be defined by {.} .

6) If any number is at least {._.}_AND_{.}, then a bijection between Q or R sets to proper subsets of themselves, can be defined if and only if any Q or R member is both some unique element, and a scale factor of the entire set (in N or Z the scale factor result is always out of the domain of the original pairs, when no one of the pairs is 1,-1 or 0).

7) My new system is consistent and well-defined, and cannot be understood by Standard point of view (yellow glasses).

8) Standard Math has no answer to the question: "What is a number"?

My theory gives rigor answer to this question by using the information concept, where redundancy and uncertainty are fundamental ("first-order") properties of it.

Standard Math uses only the on_redundancy no_uncertainty information form
as "first-order" property.

9) Furthermore, my theory includes our cognition's ability to define 'a number', as a legal part of it.

 

[CrankFan]

"If you do not understand my paper, then you cannot show us any meaningful conclusion about it" -Lama

I can conclude that the paper is incoherent babble.

If you want to formulate a new number system, you're free to do that however as Matt already pointed out if your "reals" aren't equivalent to our reals then why would we want to replace our reals with your "reals"? You've already dropped a few hints that indicate that your "reals" aren't equivalent to our reals. That you would try to pass your "reals" off as The Reals strikes me as incredibly arrogant.

Which gets to the crux of the issue. No one seems to be making a fuss about the claims you've made regarding your proposed number system, but rather the claims you've made about the number system of "standard mathematics" and how your number system is so much better than what it offers. That, and your apparent ignorance on the subject of the number system of standard mathematics.

 

[Lama]

Matt already pointed out if your "reals" aren't equivalent to our reals...

What are you talking about?

What do you mean by "our reals" ?

Do you think that I am an alien out of space?

We are all living in this beautiful blue planet and share our thoughts with each other (and now we can do it by internet).

In my work I clearly show that what is called "the Langauge of Mathematics (including its logical reasoning)" is based only on no_redundancy no_uncertainty information form (as its "first-order" property) which is nothing but some private case of infinitely many universes of different information forms.

I take my theory and try to share this deep insight with you.

What I find is a community of religious people that do not want to examine any new idea (that maybe can enrich their world) even if we still talking in the philosophical level.

In my opinion, the Langauge of Mathematics is maybe the most powerful language that we have, that maybe has a direct influence to our survival on this blue planet.

Therefore we have to be opened to new fundamental ideas about this language and put aside this stupid (and in my opinion, even a dangerous attitude) "our xxx" manifests.

Think big man, let us get together out of our defensive corners and at least open our minds to the possibilities of ideas that can be found in each one of us.

In short, do not kill things too quickly, because the first rule of survival is to find the balance between opposite things.

 

[ahrkron]

It is not a matter of politics, survival, or religious behavior.

The problem is, Lama, that you are starting from your own constructs and then trying to pass them as concepts that have a different definition, one that is standard and well agreed upon (or many definitions that can be shown to be equivalent to each other).

Also, you need to learn much more about math's concepts and the proper use of its language. Otherwise, you will continue to put your effort on a crusade that, frankly, is not producing anything worth it.

I don't mean to be rude, but you need to realize that what you are doing is not math or anything close. Not only that, it is also inconsistent, unnecessary, ugly and, to make it worse, badly written. It is not a matter of people being "rigid" or "close-minded", but of you needing to learn more and to develop skills on mathematical thinking and comminucation.

 

[Lama]

ahrkron, my heart with you, because my work has a strange property.

When you look at it, you see your own real face.

 

[Lama]

Dear ahrkron,

I do not think that your last post is the best you can do.

So please use your professional mathematical skills, and give your detailed answer to post #34.

Maybe the dialog between ex-xian (which is a moderator of another forum) and me, can help here, so here it is:

Thanks for the kind words...(this thread is becoming a mush fest ).

Please give your detailed epsilon-delta proof, which is not based on a proof by contradiction, and please add an informal explanation to each formal part of it, thank you.

The definition of a limit is as follows:
lim_x->c f(x) = L (the limit, as x tends to c, of f(x) is L) means that given an ε>0, there exists a δ>0 such that 0<|x-c|<δ implies |f(x) - L|<ε.

The point is that given any epsilon, you can always find a delta that corresponds to that epsilon so that the above implication holds. There is no magic delta that holds for all epsilon.

Here's a very simple example of a proof that the lim_x->2 2x+3 = 7, written in line form rather than paragraph form to make it easier to refer to.

1) Let ε>0
2) We seek a δ such that if 0<|x-2|<δ then |(2x+3)-7|<ε.
3) Let δ=ε/2.
4) Let |x-2|<δ, that is |x-2|<ε/2.
5) Then 2|x-2|<ε,
|2x-4|<ε, and
|(2x+3)-7|<ε.
6)That is |f(x)-L|<ε. So, by definition, lim_x->2 2x+3 = 7. qed

1) An arbitrary epsilon is "chosen." The epsilon is abitraray so no generality is lost, but in no way does it stand for every epsilon greater than zero.

2) Just a statement of what we're trying to do. For a given epsilon, there is one of more corresponding deltas that fulfill the requirments. For example, if ε=1, then a delta of 1/2 would be sufficient.

3) The delta is chosen. This is accomplished by manipulating |f(x)-L|<ε to get it in the form of |x-c|<(an expression that involves ε). In this example:
|(2x+3)-7|<ε
|2x-4|<ε
2|x-2|<ε
|x-2|<ε/2.
This gives the connection between the chosen epsilon and the delta.

4)Statement of definition.
5)Showing definition holds.
6), 7) Closing statements of proof.

Hi ex-xian,

Thank you for the very clear post about epsilon-delta proof.

It is based on a bijection (edit: insted of bijection please read injection) between infinitely many arbitrary ε to their unique δ, where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).

Well my friend, in my paper I am talking about the logical (general) meaning behind any epsilon-delta proof, which in this case uses ε and δ connection to show how any given interval |x-x0| implies δ=0.

Let us write this logical proof by contradiction:

If |a-b| = δ < all ε > 0 then δ = 0.

Proof:

Let us say that δ > 0

1) δ < all ε > 0
2) δ > 0

Since δ < all ε > 0 and d > 0 then δ<δ that cannot be true, so (1) and (2) cannot both be true.

Therefore, it is true that If (1), then not (2) --> δ = 0, QED (a proof by contradiction).

In my paper I use this proof to show that it is limited to an excluded-middle logical reasoning (exactly as there is no magic δ for all ε).

This is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:

1) Logical reasoning.

2) Limit.

3) universal quantification.

Please read again http://www.geocities.com/complementarytheory/ed.pdf , thank you.

Thank you,

Lama

 

[kaiser soze]

Lama:

Proof:

Let us say that δ > 0


Did you mean "let us say that there exists delta such that delta >0"?, otherwise your statement makes no sense, delta is not a literal.

Kaiser.

 

[matt grime]

Originally Posted by Lama


Hi ex-xian,

Thank you for the very clear post about epsilon-delta proof.

It is based on a bijection between infinitely many arbitrary ε to their unique δ
where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).



Wrong, again, about a piece of mathematics, yet you are about to state something categorically about it. Given an epslion, there is not a unique delta, in fact there need to be uncountably many possible deltas for any given epsilon, hence there is no bijective correspondence. But you've always indicated that you don't actually know what a bijection is so I suppose we shouldn't be too hard on you.

 

[Lama]

Well Matt you are well-known by your profound and quick conclusions.
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Hi ex-xian,

Thank you for the very clear post about epsilon-delta proof.

It is based on a bijection between infinitely many arbitrary ε to their unique δ
where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).

No such bijection exists. Although no delta works for every epsilon, for a given epsilon there is an infinite number of possible deltas.

Well my friend, in my paper I am talking about the logical (general) meaning behind any epsilon-delta proof, which in this case uses ε and δ connection to show how any given interval |x-x0| implies δ=0.

That's not what a delta-epsilon proof is supposed to show. I gave the formal defintion in my previous post.

Let us write this logical proof by contradiction:

If |a-b| = δ < all ε > 0 then δ = 0.

Well, again, this isn't what a delta-epsilon proof is for. Also, this is trivially true. The only non-negative number that is less than every postitive number is 0.

A delta-epsilon is used to show the existence of a limit. What limit and function are you working with?

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for a given epsilon there is an infinite number of possible deltas.

Lama:

Thank you for the correction.

By mistake I wrote bijection instead of Injection.

So, if for any given epsilon there is at least one delta, we can say that there is a 1-1 and not onto between an epsilon and a delta (this correction has no influence on my argument in this post).

...in this case we use ε and δ connection to show how any given interval |x-x0| implies δ=0.

this is trivially true

Lama:

It is trvially true according to the framework that you choose to work with.

For example, it is trivialy true if:

1) 'If |a-b| = δ < all ε > 0 then δ = 0' is an hypothesis ('If' is used).

2) A universal quantification can be realted to a collection of inifintely many elements.

Let us write again this logical proof by contradiction:

If |a-b| = δ < all ε > 0 then δ = 0.

Proof:

Let us say that δ > 0

1) δ < all ε > 0
2) δ > 0

Since δ < all ε > 0 and d > 0 then δ<δ that cannot be true, so (1) and (2) cannot both be true.

Therefore, it is true that If (1) , then not (2) --> δ = 0, QED (a proof by contradiction).



In my paper I use this proof to show that it is limited to an excluded-middle logical reasoning (exactly as there is no magic δ to all ε).

This is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:

1) Logical reasoning.

2) Limit.

3) universal quantification.

Please read again http://www.geocities.com/complementarytheory/ed.pdf

In short, what I want to show here is, that fundamental concepts of the Langauge of Mathematics, can have different interpretations in different frameworks.



You can say that you are interested only in the common framework and you don't care about any other possible framework.

If this is your basic approach, then it is ok with me, but in this case each one of us is talking to himself, and there is no dialog but two monologs.

And for monologs I do not need this thread.

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Did you mean "let us say that there exists delta such that delta >0" ?

Yes.

 

[kaiser soze]

I am sorry, but I could not find the function and the limit in your posts. Delta/Epsilon proofs are typically used in context of a given function and limit.

Kaiser.

 

[Lama]

In my paper I use this proof by contradiction to show that it is trivial only in an excluded-middle logical reasoning.

My paper is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:

1) Logical reasoning.

2) Limit.

3) universal quantification.

Please read http://www.geocities.com/complementarytheory/ed.pdf

In short, what I want to show here is, that fundamental concepts of the Langauge of Mathematics, can have different interpretations in different frameworks.

 

[kaiser soze]

Your statements make very little sense. I think you do not really understand what epsilon/delta proofs are used for.

 

[Lama]

Dear kaiser soze,

Math has many faces, and if you choose to ingore it, then we have no dialog between us.

 

[PFanalog57]

Dear kaiser soze,

Math has many faces, and if you choose to ingore it, then we have no dialog between us.

Why not re-derive the foundations of mathematics?

Assume that a universal set exists. Forget about the axiomatic enigmas generated by a "bottom-top" approach. Don't let the dogmatic zealots dampen your spirits with their flatulant barkings.

Start with a top-down approach, using an all inclusive symmetry axiom.

Symmetry forms the basis of truth.


Existence is a definition, a predication, which is why Kant so
vehemently denied that existence is a predicate, but alas, existence
is a definitive constraint as is all definitions. To exist means to have some instantiation. So infinite paradox becomes infinite freedom from definitive
constraint, and reality itself is a equilibrium point.

So by defining a largest possible set, we create a paradox,
because the set of all sets is its own power set; the
cardinality of the set of all sets must be bigger than itself along with the Bertrand Russellian set that does not shave itself. .

The "Ein Sof" is an infinity that cannot be comprehended. For every
set A there is a choice function, f, such that for any non-empty sub
set B of A, f(B) is a member of B, and so we see that there may be
an infinite number of sets B within A, and as such the Banach-Tarski
paradox is created. A single sphere is decomposed and re-assembled
into two spheres, each with the same radius as the original sphere.

So we see that:

[paradox] = not-[paradox]

is a paradox of course!

therefore:

paradox = paradox

is absolutely true.



Alpha = Omega

It is the categorical formulation of the simultaneous, situational,
instantiated contradiction, where deductive invalidity is the product
of the utmost categorical truth of the assumption that if the
antecedent of a true conditional is false, then the consequent of the
conditional is true or false indifferently, and of the categorical
falsehood of the conclusion condequently predicates that if it be not
the case that the consequent of a true conditional is true or false
indifferently, then, it is not the case that the antecedent of the
conditional is false. To pronounce the consequent of a true
conditional as being true or false indifferently is tantamount to
saying modally that where the antecedent of a true conditional is
notoriously false, then the consequent can, or could be, or is
possibly true or false. But it may be worthwhile to see that the
definitive, simultaneous equality of both true, and false, can be
formulated without explicitly including modal terms, which become the
predicating operators, which, for the sake of showing that the
consequent paradoxical conundrum is not straightforwardly resolvable
by appealing to concrete philosophical scruples concerning the
intensionality of predicated modal contexts.

The categorical representation of the propositional anti-logic, in
which deductive invalidity depends on the modality of the truth
conditionals concerning the prerequisite of the contingent assumption
and consequent conclusion. The totally relevant content of the
assumption and conclusion, definitely contains no modal terms. But,
the modality attaches to the fact that the conditional assumption is
quite possibly true, while the conditional conclusion is necessarily
false.

Which leads us to an argumentational representation of a completely
non-bogus modal formulation of the "paradox". Deductive invalidity is
most excellently predicated on the categorical truth of the
modal-term-laden assumption and the definitive categorical falsehood
of the modal-term-laden conclusion. Hence, the assumption is such,
that if the antecedent of a contingently true conditional is false,
then, the consequent of the conclusion can be true is itself quite
simply, ...true. Therefore, the conclusion that if it is not the case
that the consequent of a contingently true conditional can be true,
then it is not the case that the antecedent of the true conditional is
false, is itself quite simply, false.


Architechturally speaking:

An exact computational correspondence, i.e. "one to one and onto", becomes a type of "limit", in the compression of information via powerful generalizations. At the limit of informational compressibility, the physical universe is actually a mathematical universe. Does this limit exist? If it exists, we live in a mathematically designed universe, governed by perfectly harmonious equations. In fact, the design and construction phase are an approach towards an equilibrium point.

Ideal mathematical perfection, creates problems for itself. These problems arise with the introduction of "free will" and sentience to the equational composition. Systemic anomalies[rebellious sentient programs] rear their ugly heads. Yet, without sentient beings possessing the attributes of free will and the ability to make a "choice", the universal system of progressively compositionial calculations could not approach the limit of infinite informational equilibrium, and the system would disintegrate.

Counterbalancing occurs with the sentient "Heisenberg compensation" operators, which must be introduced, to maintain the most optimal trajectory towards a state of perfect equilibrium with maximal efficiency. Nomological covariance and consistent history, is maintained in place of an absolute deterministic exhaustion. A totally closed system. Thus these "guardian" sentient programs must police the timeline, ensuring that it remains paradox free.

A recombinatorial mix of sentient attributes allows for further maintenance of an optimal equational trajectory, as sentinel "messiah programs" are martyred in the war against the rebellious "systemic programming anomalies" .

 

[Lama]

Don't let the dogmatic zealots dampen your spirits with their flatulant barkings.

Dear Russell E. Rierson, thank you, but I want to add that I have no problem with dogmatic approach of others, if I can use it to develop my work.

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.

So we see that:

[paradox] = not-[paradox]

is a paradox of course!

It depends on the framework that we choose to work with.

In an excluded-middle reasoning a = not_a is nothing but a false statement.

 

[PFanalog57]

Dear Russell E. Rierson, thank you, but I want to add that I have no problem with dogmatic approach of others, if I can use it to develop my work.

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.

It depends on the framework that we choose to work with.

In an excluded-middle reasoning a = not_a is nothing but a false statement.

contradiction = not-contradiction is a contradiction

true?

or

false?

 

[Lama]

There is no question here.

a is not_a is nothing but a false statement in boolean logic, because no identity can be in more than one unique state in boolean logic.

 

[matt grime]

when did "is" become a connective in boolean logic?

do you mean "and", which is after all the definition of a paradox, something that is simultaneously true and not true.

 

[Lama]

By the word 'is' I mean to '='

a = not_a is nothing but a false statmant in boolean logic.

 

[Lama]

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.

 

[PFanalog57]

There is no question here.

a is not_a is nothing but a false statement in boolean logic, because no identity can be in more than one unique state in boolean logic.

If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?

 

[Lama]

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.

 

[matt grime]

eh? so A iff not A is false, so? (note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic) what does that have to do with anything? what matters is that if we adopt naive set theory we have a case where A and notA must be true, when it is trivially false, so what?

 

[Lama]

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 statement 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.

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.

Also read post #48.

 

[Lama]

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 referred 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.

 

[Lama]

Some dialog:

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Lama:

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?

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kaiser:

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.

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Lama:

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).

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kaiser:

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,...

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Lama:

-------post #190

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.

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Lama:

-------post #191

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 (any transformation from A state to B state cannot be but a quantum-like leap).

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.

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Lama:

-------post #192

'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.

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kaiser:

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.

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Lama:

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, which can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

 

[matt grime]

"In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. "

once more you demonstrate you ignorance of mathematics. take an eventually constant sequence to see why.