Closed Intervals with Infinite Endpoints: Explained

[Organic]

Hi phoenixthoth,


I try to explain a system which is multi-dimensional by nature, to persons who insist to translate it two 2 dimensional system.

Another example, I try to explain a colorful system by nature, to persons who insist to translate it two black and white system (Boolean Logic) Or greyscale system (Fuzzy Logic).

It simply can't be done.

What you write simply show me that I did not succeed to explain what is Complementary Logic.

Complementary Logic is first of all a paradigm changing in the question: "what is Mathematics?".

I am not talking here about some technical improvement, but on something that is changing math from its conceptual fundamental level, no less no more.

If you still trying to look at Complementary Logic through the Boolean or Fuzzy Logic eyes, then you cannot understand even one thing in Complementary Logic.

I hardly tried to open your eyes to Complementary Logic, but from your last post I realize that I did not succeed yet.

You still trying to find it under the spotlight of Boolean Logic and Euclidean Mathematics.

So, let me simply tell you that you will not find it there, again because we are dealing here with a paradigm change, no more, no less.


Yours,

Organic

 

[phoenixthoth]

i would guess that category theory is also a paradigm change, to give you an example, an abandoning of sets as the fundamental object. if you abandon logic, even fuzzy logic, then you can no longer use deduction:
(A&(A-->B)) --> B,
because that's a statement in binary logic, yet you still use deduction don't you? hence you're using binary logic to escape binary logic. this can be done with some delicacy. but to be a mathematical theory, there still needs to be definitions of terms (even category theory has definitions), a set as small as possible of undefined terms (preferably empty), and "rigor." one commonality in your articles is a very rapid jump signaled by your use of the word "therefore." this is not automatically a bad thing, but i would say that as it is, it is too rapid. several of your "therefores" are "non-sequitors," which means one of two things:
1. i don't understand the logic
2. the conclusion just don't follow from the premises.

keep trying and i will work with you to parse out the nonrigor and sift through it. but you have to at least try to incorperate my advice when i point to something.

just let me get one thing straight: what is your primary goal in five sentences or fewer? i can't argue with your goals.

now, after your primary goal has been stated, your thesis statement, give me an outline, not intended to prove it, of how you will accomplish your primary goal as briefly as you can.

we will talk about that.

then, and only then in my opinion, should we talk about the details and how to go about doing it. at that point, if and when we get there, give me small spurts of things to consider, rather than a ton of articles. if i want to referee your work, i will ask you a question about the first thing that seems incorrect or unclear; so it wouldn't help to send me pages of math if i will have a question with the second line. i think this is how we should carry on from this point. in other words, I've read your articles already so sending them again and saying "don't you see?" will not help move things foward. for now, just state your primary goal which has to do with noneuclidean mathematics, not to be confused with noneuclidean geometry i realize, and a very basic outline of how you will achieve that primary goal.

let me give you an example of what i want:
in my ternary universal set theory article, my primary goal is to axiomatize the universal set, the set of all sets, into existence.

that's it. that's my primary goal. one sentence.

steps:
1. use ternary logic to
2. extend the subsets and foundation axioms so that i can
3. axiomatize U into existence and
4. remove the usual problem, russell's paradox while
5. showing where the normal theory wouldn't apply with the extended subsets axiom.

this is all i want for now. no details. this will help me grasp what you are trying to do. then, after you have stated your primary goal and basic steps, we will go over the details line by line and work towards a rigorous theory.

 

[Organic]

My goal is to find a theory that can associate between at list two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.

 

[phoenixthoth]

ok. thank you.

now i want to really understand what you're trying to say:

My goal is to find a theory that can associate between at list two opposites.

english question: do you mean "My goal is to find a theory that can associate between a list two opposite descriptions or characterisitics?"

by "list," do you mean "set" or not? for example, if C and D are two categories, do you want to apply your goal to the list {C,D} or only when C and D are sets or something else? and by "opposite," do you also mean "complimentary?"

once you answer these questions, also i need to point out that "by using the simplest possible means" is actually a part of the goal, i think. in other words, it's not an outline of the steps you will take to assiciate a list with two opposites. i don't want details yet, just some outline more specific than "by simplest possible means" and more general than all the details.

a side question not of much importance now is this: do you intend to prove that your means are the simplest? i ask because that could be very difficult to do.

also, can you give me an idea as to what kind of two opposites you mean and in what sense they are opposites?

and, finally, the more you give me, the more i or someone might be able to tell you if this, or something similar, has been done before. if something similar has been done, it will help you immensely to become familiar with it and go where it does not.

 

[Organic]

Forgive me about my English, the right one is:

My goal is to find a theory that can associate between at least two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.

 

[phoenixthoth]

what kinds of opposites?

opposite numbers?

opposite sets?

opposite elements in a group? (group theory is a fairly general setting for looking at structures with opposites.)

something else?

in category theory, there is a concept of opposite category...

 

[Organic]

Dear phoenixthoth,

Please read all of this, thank you:

My goal is to find a theory that can associate between at least two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.


Step 1:
The first thing is to find the most general concept to start with, so we choose information.


Step 2:
Then we choose the limits of any information system, which can be defined as at least to opposites, so we choose No information, Total Information.


Step 3:
we are useing these limits as the contents of two opposite set's types, where the set's idea is a tool that we call it General Information Framework(GIF), which is the model or the platform that we use to explore our ideas.
http://www.geocities.com/complementarytheory/GIF.pdf


Step 4:
Now we look for simplicity by using the symmetry concept as the balance between {} and {__}.

{} and {__} are the unreachable limits of our system, which is a fading transitions between these limits, and only the products of the fading transitions can be explored as meaningful Information. By using the open interval idea the meaningful information exists in ({},{__}).


Step 5:
The first symmetry break is a model of infinitely many empty information cells existing upon infinitely many scales, where cells size expending (aspirating to) {__} an shrinking (aspirating to) {}.

The second symmetry break is to "left-right|right-left" symmetry by fill the empty information cells with the minimum necessary information that can break the symmetry.
http://www.geocities.com/complementarytheory/LIM.pdf

The third symmetry break is the floating point system that splitting the Information cells to two opposite directions, integer and fractional.

By using Riemann's Ball we find the full symmetry between integer an fractional sides.
http://www.geocities.com/complementarytheory/RiemannsBall.pdf

Also By using Riemann's Ball we find the difference between actual infinity and potential infinity.

Also we find that potential infinity can never be completed and this property do not give us any possibility to use the words 'all' or 'complete' when we explore it.
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf


Step 6:
With this knowledge in our hands, we realize that The Number System of Standard Mathematics is some arbitrary broken symmetry between integer and fractional sides, where fractional side is full, but the integer side includes only numbers with finite length.


Step 7:
At this stage we stop continuing are main program to show the problems that we have found in Standard Mathematics from our new point of view.

The problems of Standard Math that we have found:

1) It is not aware that it is based on some arbitrary broken symmetry between its integer side and its fractional side.

2) It does not distinguish between potential infinity and actual infinity, and therefore using words like 'all' and 'complete' related to potential infinity.

The result of this mistake is the transfinite universes, which is nothing but a "full gas in neutral".

3) It is based on very week methods like Boolean Logic (black XOR white system) OR Fuzzy Logic (Grayscale system).

4) Standard Math is based on the quantity concept, therefore a lot of very interesting information are out of its scope.

5) There is no difference between multiplication and addition.

7) There is no general definition to the Number concept.

8) Concepts like redundancy and uncertainty are not fundamental concepts.


Step 8:
We continue our main program to find the logic system that will be the base of our system. The result is what we call Complementary Logic.
http://www.geocities.com/complementarytheory/CompLogic.pdf
http://www.geocities.com/complementarytheory/4BPM.pdf


Step 9:
By using Complementary Logic, we reexamine the concept of The Number and starting to make the first general sketches of Complementary Logic Number System.
http://www.geocities.com/complementarytheory/AHA.pdf
http://www.geocities.com/complementarytheory/Everything.pdf
http://www.geocities.com/complementarytheory/ASPIRATING.pdf
http://www.geocities.com/complementarytheory/ET.pdf
http://www.geocities.com/complementarytheory/CATheory.pdf
In these sketches we can clearly show that Complementary Logic is based on stapes 1 to 5 and fix the problems that have been found in steps 6 and 7.

We also think that Complementary Logic can be very useful in Quantum Mechanics (the micro level) and also it can be used as a very good basis for modals that dealing with static an dynamic complexity (in mid and macro levels).

The reason that we think so, is because Complementary Logic using in a coherent way concepts like Information's clarity-degree, Symmetry-degree, redundancy, uncertainty, Information structure and quantity.


Because Complementary Logic is a "colorful" system, we try to explore its frontiers by checking subjects like "our abilities to count" and more subjects that are connected to our own cybernetic systems.
http://www.geocities.com/complementarytheory/count.pdf
http://www.geocities.com/complementarytheory/RealModel.pdf
http://www.geocities.com/complementarytheory/CK.pdf


Step 10:
We examine the connections between Complementary Logic ,Moral and Art.
http://www.geocities.com/complementarytheory/Moral.pdf
http://www.geocities.com/complementarytheory/O-Harp.pdf


Step 11:
We hope for some help.
http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf



Yours,

Organic

 

[phoenixthoth]

since this is a theory about information in general, it is, strictly speaking, not a mathematical theory but a theory more general than a mathematical theory.

i don't really know much about researching nonmathematical theories but i would imagine the first thing you should do is define information and specify the scope of your theory. is it going to apply to all kinds of information?

you may want to run a search for "information theory."

 

[Organic]

Please don't say that.

Dont you realize that 'Information theory' of today is a MAthematical theory?

Did you read steps 6 and 7?

If you read them then how can you say that my goal is not also going through Math?

 

[phoenixthoth]

i know very little about information theory. i think that generally, general information should be more general than math and though it can be applied to any kind of information, including math, it's not just a math theory. it can use math in it, but if it's about information in general, then it may not be a math theory. either way, you haven't addressed the other point i wrote about which was that in order to talk about information, you have to either say what its definition is or why we should allow it to remain undefined.

 

[Organic]

What is undefined?

 

[phoenixthoth]

if I'm not mistaken, sets and points from geometry are undefined. in other contexts, points are defined.

 

[Organic]

Dear phoenixthoth,



In all of what i wrote, please show me what is undefined and must be defined?

Thank you.

Yours,

Orgainc

 

[phoenixthoth]

information, for one thing.

 

[Organic]

Why do I have to define it?

 

[phoenixthoth]

it is listed in step 1 as the object of study.

you may not have to define it but there should be several examples of things that are information and things that aren't.

set theory is a theory about an undefined concept but one can say something like, here are four "widgets" and here are seven ways to build new "widgets" from old ones, but i won't tell you what a "widget" is. that's set theory, at least. maybe you can follow a parallel structure in information theory.

if this is to be a mathematical theory, i think you'd have to decide on a set of constants like ∈ though ∈ doesn't have to be one of them.

what would be nice is if you could fit information theory into an existing theory so you get to use all of its power. information itself seems to be more general than even logic and in fact logic would be a subject of study in information, as would illogic. they're both information. it is ok to use logic to study logic or to use logic to study illogic if you do it delicately.

the examples I'm keeping in the back of my mind is how the definition should include the following information:
1. information about what a set contains
2. information about how I'm emotionally feeling
3. information of a poem
4. computerized information
5. information kernels, ie, truly abstract information
6. nonverbal and nonwritten information
7. the relationship between information and truth (eg true information)

so i think that if this is going to be about information in general, it should capture all kinds of information. if information is undefined in terms of standard math words, it will take a lot of "motivation" for anyone to know it. in other words, what will be the major theorems? give at most one for now without proof.

 

[Organic]

Dear phoenixthoth,



The major theorem is very simple:

No model of x is x itself, that's all.

To any development of x there is some meaning only in the gap between x-model and x.

Now, x can be Information, Mathematics, and so on.

Shortly speaking, x has two basic forms: x-model, x.

The problem of any research is not to forget the above during the research.

Now let us call x-model potential x, and let us call x actual x.

Modern Math language forgot this and the result is the transfinite universes.

Another importent reason to this result is:

Modern-Math Number-Systems are based on some arbitrary broken symmetry.

To see it, please look again at:

http://www.geocities.com/complementarytheory/LIM.pdf



Modern Math in general does not distinguish between x-model and x.
Therefore it becomes a closed and circular system.

Take for example your comment about Math:

Since this is a theory about information in general, it is, strictly speaking, not a mathematical theory but a theory more general than a mathematical theory.

My response to this is:

There is no such a thing "mathematical theory" because any theory can be only
x-model, and no x-model is x.


Conclusion: Any x-model is an open system that can be changed.

Please read again both of them:

http://www.geocities.com/complementarytheory/GIF.pdf

http://www.geocities.com/complementarytheory/RealModel.pdf



Yours,

Organic

 

[Hurkyl]

Why do I have to define it?

So you can apply logic.


At the very least, you have to enumerate the basic facts about things that allow us to start proving theorems.

For example, ZFC doesn't even try to say what a set is, but it rigorously lists the operations we're allowed to do on sets (e.g. make a pair set, make a power set, make a sumset, make a subset), thus allowing to prove theorems, et cetera.

Euclidean Geometry doesn't try to say what point, line, between, incident, or congruent is, but it precisely lists some facts about them (e.g. for any two distinct points there is a unique line incident with both), thus allowing us to rigorously prove theorems from these basic facts.

 

[russ_watters]

Originally posted by master_coda
I'm well aware of this. But I don't really take it serious enough to get frustrated over it.

Well, you're a glutton for punishment with bonus points for tenacity. Good luck!

 

[Organic]

Hi Hurkyl,

Please read my post (with the steps) and also my post about x-model and x, and see for yourself how Information concept is extremely productive concept in my work.

If you don't think so, please tell me why?


I'd like to know what do you have to say about Complementary Logic:

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/4BPM.pdf

 

[phoenixthoth]

ok, organic. you've stated your goals and your steps and your main theorem. so far, so good. this would be quite an undertaking.

as for your main theorem, what is x? is it an information "system", a set of information, etc.?

you might find this idea interesting. the claim is that the universe contains almost no information:
http://www.hep.upenn.edu/~max/nihilo.html

i already think that (almost) no model of x is x. I'm just taking for example a model for gravity and gravity. what about metamathematics (model theory, set theory, logic)? it is a model for mathematics and it is (part of) mathematics.

 

[Organic]

Any Model is on x and never the x.

For example: To eat the cake is the x, but to speak on eating the cake is a model of eating the cake (a x-model).

In mathematics "Eeting the cake" = "Actual infinity".

Shortly speaking, no theory can deal with Actual infinity, but can use a model of it, which is potential infinity.

Also, the main player on this stage is first of all the symmetry concept.

 

[phoenixthoth]

i think there can be a finite model of absolute infinity. no one is saying that the model for infinity is infinity i don't think.

now, before you launch into a discussion on that, you still haven't really integrated our feedback into what you're doing, which makes us giving feedback less purposeful. and the feedback was this: define information or leave it undefined but describe examples of it and gives ways to contruct new information from old information as how it is done in geometry and set theory.

 

[Organic]

I already did it, please read pages 7,8,9,10 (The introduction):

http://www.geocities.com/complementarytheory/CATheory.pdf

 

[phoenixthoth]

i did a text search for the word "information" and didn't see a definition or a scheme of what is or is not information. therefore, the reader does not really know what your theory applies to. we all have a kind of intuitive sense of what information is but in order to be considered a mathemtical theory, or a philosophy of mathematics theory, information has to be defined or at least how to get new information from old information (with a collection of what some information is) has to be done. otherwise, it will be impossible to prove any statement about information in a rigorous way.

maybe I'm just missing it. if you did this, please point out the specific page number and line number where you define or give an inductive definition of information. i see the word information used on the pages you listed but nothing resembling a definition. a definition or inductive definition is necessary in order to prove anything about information in a mathematical theory. you may want to start with the dictionary definition of information and try to turn that into a mathematical definition. however, the words used in your definition must also be defined or inductively defined. you may also want to look at information theory and see how they define it.
http://en.wikipedia.org/wiki/Information_theory
http://en.wikipedia.org/wiki/Information

by "inductively defined," i mean that if you don't define information, or words used in your definition of information, you should give a few examples of information and then give a list of ways to construct new information from old information. if you use other words, like entropy or information clarity degree, or symmetry, then the same applies to those words because they have either no common definition or a definition that depends on context.

why do you have to define (inductively or not) words?
1. in order to prove something about those words
2. if not, you run the risk of "abusing language" such as but not limited to changing the definition or implied meaning of words in mid article or even mid sentence.

 

[Organic]

My Definitions are given by the structures themselves.

The words and the sentences around them just giving an extra explanations
to what is already given by structures.

Shortly speaking, my definitions are "structure oriented".

If you understand this, then look again at http://www.geocities.com/complementarytheory/CATheory.pdf starting from page 7 until the end, thank you.

The beauty in my theory is: the structures are the definitions and the examples.

 

[Organic]

A definition for information:

A product of a mutual influence between, at least, two different things.

 

[phoenixthoth]

Originally posted by Organic
My Definitions are given by the structures themselves.

The words and sentences around them just giving an extra explanations
to what is already given by structures.

Shortly speaking, my definitions are "structured oriented".

If you understand this, then look again at pages 7 and this time go until the end, thank you.

The beauty in my theory is: the structures are the definitions and the examples.

it sounds like you're trying to get away with defining in your article information by context. in other words, the term is meant to be "defined" by the words around it. this isn't a rigorous definition. the one you just gave is closer to an actual definition.

is the sentence, "yes" an example of information?

how about the formula (x->(y<-> ? it's not a "well formed formula" but would it be considered "information" because it is about five different things? btw, what is a thing? a set? a letter? a symbol?

under your definition, the sentence "yes" is not information whereas "(x->(y<->" is information. i just want to clarify what information is. there is no such thing as a wrong definition; only good and useful definitions or bad and useless definitions.

i can see how "x is x" is information because it is about two differnent things: x and is.

seems that there are at least two kinds of information: sensical information and nonsensical information. there would probably also be degrees in between which suggests an application of a fuzzy approach. there could be a "sense indicator" S so that if x is information then S is a map from the collection of all information to [0,1] such that S(x) is in [0,1] and S(x)=0 means that x is devoid of any sense (perhaps this is total entropy) and S(x)=1 means x is totally sensical (perhaps this is total negative entropy). then you can develop some conditions on what kinds of S's are actual sense indicators because something that makes sense from one perspective may not make sense from another perspective.

the sensical indicator would have nothing to do with the truth of the information, it would just measure how "grammatically correct" the information is.

seems like there should be a definition of "more information" and "less information." a kind of relation between different information resembling subsets and supersets.

my main point here is to just say that defining a word by context simply won't do in a rigorous theory. however, your recent definition is much better. in order to be a mathematical theory, you should define what kinds of things you're considering. a thing is perhaps as general as you can get and goes way beyond math (unless mathematical existence is physical existence, that is).

you'll need a definition of "product" and "mutual influence" where those words don't depend on the definition of "information."

i want to reiterate that i already believe your main theorem without any work: no model of a system is the system, so you may not have to go through all the trouble you're going through.

however, from that, you conclude radical claims about transfinite objects. firstly, it is totally unclear how that follows from "no model of x is x" and secondly, in order to really convince anyone that 150+ years of set theory is wrong, you have to show where the error is. these are short and long term goals, respectively. for now, please just speak about my questions for clarification of what information is and is not, what kinds of "things" you're talking about, what "product" means, and what "mutually influence" means. my main question about "mutually influence" is that i don't see how saying, "the force of gravity acting on two masses M and m is given by the function F(M,m)," shows an influence between the force of gravity and the formula itself. the influence you must be talking about is on some kind of linguistic level because the formula does not influence what the formula refers to (emperically speaking).

to shorten this down for you, maybe just talk about the following things in your next post:
1. is "(x->(y<->" information? (there is no right or wrong answer here)
2. what is "product"
3. what "things" are you talking about (the word "anything" should be used delicately here)
4. what is "mutually influence"

thanks

 

[Organic]

"One picture = 1000 words"

Don't you see how rigorous are my structures?

you'll need a definition of "product" and "mutual influence" where those words don't depend on the definition of "information."

I already gave an example for this in page 8 of http://www.geocities.com/complementarytheory/CATheory.pdf

 

[Organic]

...the formula does not influence what the formula refers to ...

In any formula is x-model, the infuence is by x.

SASs can link between x-model and x.

Maybe this is the most fundamental SASs property.

 

[Organic]

About the transfinite numbers, I already gave you my answer.

I'll write it again, but this time i use |Q| instead of |N|.

Rseq is actually both R and Q.

( http://www.geocities.com/complementarytheory/NewDiagonalView.pdf }

The way Rseq is constructed is equivalent to both |Q| and 2^|Q| (or |P(Q)|).

This is the reason why we get this result (2^aleph0>=aleph0)={}

Form one hand Rseq is |P(Q)|( =[...000,...111) ).

From the other hand Rseq is |Q| ( = The length of each given sequence ).

Please tell me why it is so hard for you to understand the above?

Let us say it again:

Cantor's diagonal fails because he deals with the wrong input, which is |Q|*|Q|.

By the way Rseq is constructed, for the first time since Cantor we deal with the right input, which is |P(Q)|*|Q|.

By doing this we find that (2^aleph0>=aleph0)={}.

Therefore transfinite universes do not hold.

Again, Rseq is both R AND Q.

More then that:

If Rseq is [...000,...111] then it means that Cantor's diagonal input (which is ...000) does not exist.

Therefore no input --> no output --> no any information to establish the transfinite universes.

More then thet:

|P(Q)| exists iff P(Q)=[...000,...111)

Therefore there is no such a thing like all (or complete) infinitely
many objects.

And when there is no such a thing, transfinite universes do not hold.

Again, |Q| is a "never ending story", therefore words like 'all' or 'complete' cannot be related to |Q|.

 

[phoenixthoth]

one thing that had me confused is that when you meant page z, i mistook that for page z of the document and not the page with that z listed at the bottom. so i will go through it again.

Cantor's diagonal fails because he deals with the wrong input, which is |Q|*|Q|.

actually, cantor's diagonal argument doesn't use |Q|*|Q| as input. it uses any set. so |P(Q*Q)|>|Q*Q|, for example.

 

[Organic]

Thank you for your correction, but it does not have any influence on my argument that (2^aleph0 >= aleph0) = {}.

 

[Organic]

Another interesting thing is the hierarchy of dependency of R in Q, and Q in N.

Please look at this example: http://www.geocities.com/complementarytheory/UPPs.pdf

These Unique Periodic Patterns are prime-like patterns,where any irrational number uses as its building-blocks.

This example perfectly fits my argument about the power of existence that can be found in the second part of this paper (please start from screen 5 of acrobat viewer): http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

 

[Organic]

More general.

Because any mathematical system is only an x-model (therefore an open system) it cannot talk about proofs, because they’re always can be changed (or even replaced) during paradigm’s changes.

When we have a paradigm’s change, a lot of old paradigm's results can become irrelevant.

Therefore, in my opinion, Math language has to use the words 'Current Result' (CuRe) instead of 'proof'.

Please look at this nice article: http://faculty.juniata.edu/esch/neatstuff/truth.html


By using strong words like 'proof', there is (in my opinion) a danger that we become scholastic and closed systems.

And closed systems find their death by entropy.

 

[Organic]

Here you can find an example that shows why Cantor’s Diagonalization method must deal with the lows of probability.

http://www.geocities.com/complementarytheory/PTree.pdf

 

[matt grime]

Originally posted by Organic
When we find a 1-1 map between some point x to some R number, then if x in R then for any x in R, we can find some x0 < x OR some x < x2.

Therefore x0 OR x2 are always unreachable for any given x.

Let x0 be -oo(= inifinitely many objects < x).

Let x2 be oo(= inifinitely many objects > x).

No given x can reach x0 or x2.

Therefore x0 OR x2 must be the unreachable limits of any R number.

(x0,x] OR [x,x2), therefore [a,oo) OR (-oo,b] cannot be but half closed intervals.

Therefore the set of all R numbers (where R has a form of infinitely many objects) does not exist.


Shortly speaking, infinitely many objects cannot be related with the word all.

Fore clearer picture please look at:

http://www.geocities.com/complementarytheory/SPI.pdf

What does it mean to find a 1-1 map between some x and some R number? Any function mapping a single element set to R is necessarily injective, so it is either a redundant statement or it has some more meaning you aren't explaining.

next there is this x0 and x1 are not reachable by x. How does one 'reach' a number from another?

then x0 becomes -oo which then becomes a set of numbers. Do you still not understand why this needs rewriting? How can A number be -oo and be A set of infinitely many elements?

ANd in what way does it imply the Reals don't exist? Do you understand their construction as limits of cauchy sequences?

You've still not explained your private definition of the word all that means it is not realistic to talk about the set of all real numbers. give me a real number not in the set R.

matt

 

[Organic]

Dear matt,

You take some of my posts in the beginning of this thread, but you first have to read what happened since this post, because maybe your questions have been answered in the next posts.

Please check it, thank you.

 

[matt grime]

Originally posted by Organic
Dear matt,

You take some of my posts in the beginning of this thread, but you first have to read what happened since this post, because maybe your questions have been answered in the next posts.

Please check it, thank you.

Yes, some of the questions were asked previously. None of your answers have helped illuminate the issue though.

In particular you still insist that by taking the finite strings of 01s and 'completing using the infinity axiom of induction' that you get a set in bijection with N via binary expansions of integers AND a set that contains all strings of 01's. This is patently wrong.

1. Only strings with a finite number of non-zero entres will be mapped to an integer.

2. There are strings with infinitely many non-zero entries.

3. The list has been proven by you to be 'not complete'

4. Induction doesn't allow us to do what you did, inparticular the existence of an inductive set doesn't allow us to construct a set of uncountable cardinality. It just asssures us that an infinite set will exist in our model. It is countable. Then we construct more infinite sets that are of strictly greater cardinality, but not by induction. You can do transfinite induction if you so wish, but I think we've had enough without abusing the axiom of choice as well.

5. Moreover you don't even define the inductive process that tells you how to add in the next successor element.

6. Repeating myself, but, you've demonstrated no countable, enumerable, listable, whatever, set of strings of 01's contains all of them. That is sufficient, albeit that your proof could do with a lot of tidying up (the infinite case does not follow from the finite case by induction!). But you then say the list you've made is all of them anyway. Can you really not see that that is a contradiction in itself? The error is not in maths, but in your assertion you've got a complete list.

Spurious example:

1 is the largest integer. Suppose N is the largest integer, N>1 obiviously, therefore N.N>N

is a contradiction unless N =1. Do you see where that went wrong or have i given you more ammunition?

You make an unsubstantiated (indeed incorrect) claim, deduce a contradiction, but conclude it was something else that was incorrect!

Matt

 

[Organic]

Sorry but what is N.N>N ?

All what I clime is very simple: we cannot deal with x-itself, but only with
x-model, where x is infinity.

for example:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

 

[matt grime]

Originally posted by Organic
Sorry but what is N.N>N ?

All what I clime is very simple: we cannot deal with x-itself, but only with
x-model, where x is infinity.

for example:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Ok. N^2>N

The analogy is that in your New Diagonal Pdf you claim a contradiction based upon an assumption (that you have a complete list of 01 strings in bijection with N), and instead of deciding this assumption is incorrect (it is) that it is in fact the whole of Boolean logic which is at fault.

Above in the spurious example, the false assumption is that there is a largest natural number, not that 'it' is greater than 1. OK?

what is an infinity model? And if you post another pdf link can you seriously expect people to go and read it?

 

[Organic]

Matt,

Here you can find an example that shows why Cantor’s Diagonalization method must deal with the lows of probability.

http://www.geocities.com/complementarytheory/PTree.pdf

You look at the picture but don't read what I write about it.

 

[matt grime]

Against my better judgement I looked at the PTree thing.

What are yuo trying to say with it? Why must Cantor deal's argument deal with probability? In what fuzzy world are you thinking?

What is the Boolean tree of 01 sequences?

None of the sentences in the article are coherent as mathematical statements, and few are coherent as pieces of English.

It does not answer any of the criticisms of your argument about Cantor's Proof.

Simply put:

You must prove the assertion that the string of 0's and 1's that you construct 'using the axiom of infinity induction' contains all the strings of 0's and 1's. My assertion is that it does not as CAntor's proof shows, and as you yourself state (with a falacious proof). Further more you have not explained how the induction even works.

 

[Organic]

There is no such thing like "a collection of all 01 sequences".

Because:

By using the word all, we are forcing |N|(=aleph0) to be the cardinal of all N collection.

By forcing the word 'all' on a collection on infinitely many objects, we come to contradiction.

The reason is:

Cantor's diagonal is already in the collection of 2^aleph0 infinitely many sequences, because it cannot cover the collection, so we must not add it to the collection.


Conclusion 1:

2^aleph0 > aleph0 because the diagonal cannot cover the collection.

Conclusion 2:

But because it cannot cover the collection he must be somewhere in the collection, therefore we must not add it to the collection.

But because nothing is added the collection we can find a bijection between 2^aleph0 and aleph0, and we come to contradiction.

It means that we can't force the word all on any collection of infinitely many objects.

There is no such a thing like a complete collection of infinitely many objects.

 

[master_coda]

Originally posted by Organic
But because it cannot cover the all collection he must be somewhere in the collection, therefore we must not add it to the collection.

Why not? Even if we add it, the collection will still not be complete. We can always find another element not in the collection.

 

[Organic]

Master_coda,

please read again:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and see how I construct this collection of infinitely many objects.

 

[master_coda]

The fact that your constuction fails just means that your construction was bad. It doesn't mean that other constructions must also fail.

 

[Organic]

Dont you see that by using ZF axiom of infinity on the power value of 2^x, x=aleph0 by standard math notation?

 

[master_coda]

You do realize that the axiom of infinity isn't an axiom that you can "use" on things. It's just a statement that the set of natural numbers exists. Nothing more.

 

[Organic]

And without it aleph0 cannot be defined.

Therefore x (by standard math notations) cannot be but aleph0.

Therefore by standard math 2^x (where x is based on ZF axiom of infinity) cannot be but a collection of 2^aleph0 objects.

We must not ignore the meaning of the word infinite, which is "no finite" or "no end" (or "endless").

Therefore no collection of infinitely many objects can be a complete collection, because its fundamental property is not to include its end.

Because any infinite collection of infinitely many objects has no end, its cardinality is under the lows of probability.

And what is the base of this probability?

The base of the probability is first of all the value of base value n of n^aleph0.

This probability clearly can be shown here:

http://www.geocities.com/complementarytheory/PTree.pdf

Also the basic result of this probability can be shown as a complementary association between multiplication and addition (please look here):

http://www.geocities.com/complementarytheory/ASPIRATING.pdf


You say:

Before you attempt to beat the odds, be sure you can survive the odds beating you.

I say:

Before you attempt to explore the odds, be aware to the odds within you.