A question on intervals [Archive]

master_coda

Jan8-04, 03:01 PM

Originally posted by phoenixthoth
i'm taking {__} to be for now the class of all sets or a universal object.

translations:
1) a set with elements will never be the empty set

2) a set not having all elements will never reach {__}.

so, master_coda, can you explain how nothing has anything to do with math?

Well if all you have to do to talk about math is tie random math words together, then I would have to concede that what Organic says has something to do with math. Still, I would argue that he hasn't said anything meaningful or relevant.

His argument seems to boil down to "you can't talk about infinite sets because you can't count to infinity". Not only is that wrong, but it also portrays a profound lack of understanding of the mathematical concept of infinity. Or even an understanding of logic, since he asserts that "for all" is not a valid quantifier.

Organic asked a question about intervals. We posted a perfectly good answer to his question. Then we were told that our explaination was wrong. Apparently, when you change the definitions for everything, you get different results. Since we aren't told what those new definitions are, we can't even check these new results. It hardly matters anyway, since proving something about a different definition of "closed interval" doesn't prove anything about the original definition.

Hurkyl

Jan8-04, 05:06 PM

(For some reason I couldn't post this morning, but here's what I was gonna write)

It seems you're confused about what x0 and x2 are supposed to be; when you first use them:

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.

They seem like they're supposed to be real numbers.

However, when you next use them:

Let x0 be -oo...

Let x2 be oo...

They seem like they're supposed to be extended real numbers.

However...

Let x0 ... = inifinitely many objects < x.

Let x2 ... = inifinitely many objects > x.

Now x0 and x2 are sets!

(x0,x] OR [x,x2)

And now they're back to real numbers again (or maybe extended real numbers).

Make up your mind, which is it?

cannot be but half closed intervals

That doesn't prevent them from being closed sets. (and thus closed intervals)

Heck, subsets of the real line can be both open and closed simultaneously! (in particular, \varnothing and [/itex](-\infty, \infty)[/itex] have this property of being "clopen")

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

This doesn't even have anything to do with the previous statements!! Why do you say this?

Therefore definitions like 'the complete list of N numbers' is meaningless.

In your theory, maybe, but it is not meaningless in ordinary mathematical structures.

Organic

Jan8-04, 06:18 PM

The contents of {} and {__} are total states that cannot be explored by any information system, including Math Language.

Please read this: http://www.geocities.com/complementarytheory/MathLimits.pdf

Therefore any information system is limited to ({},{__}) where:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Question: Is Universal set = {__} ?

Answer:

Universal set is
the balance of ({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

To understand this, Please read:

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

The 'Boolean Logic' which is an Euoclidian-Mthematics system is goning to be replaced by systems like 'Complementary Logic' which is a Non-Euoclidian-Mathematics system.

For example read this: http://www.math.rutgers.edu/~zeilberg/Opinion43.html

Complementary Logic ( http://www.geocities.com/complementarytheory/CompLogic.pdf and http://www.geocities.com/complementarytheory/4BPM.pdf ) goes beyond the above article.

Do you still do not realize that the Cantorian world is based on a private case of some broken symmetry?

Please take a long look at:

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

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

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

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

And if you understand the above, please take a long look at:

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

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

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

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

And is you understand the above than please take a long look at:

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

http://www.geocities.com/complementarytheory/O-Harp.pdf

Yours,

Organic

Organic

Jan8-04, 06:27 PM

Hi Hurkyl,

You are right, I don't no how to use standard notations, so to make my idea clear x0 and x2 stand for infinitely many R numbers where:

x0 < some given x where x is any arbitrary R subset.

x2 > some given x where x is any arbitrary R subset.

The arbitrary R subset is represented by 01 infinitely long sequence,
taken from set Rseq, which its content = [...000,...111)XOR(...000,...111]

Rseq cunstructed by:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
Therefore it can't be "clopen".

About "clopen" I have found this:

http://66.102.11.104/search?q=cache:P3UUteuXC9EJ:planetmath.org/encyclopedia/ClopenSubset.html+math+clopen&hl=en&ie=UTF-8

Can you please explain "clopen" in a non-formal way?

phoenixthoth

Jan8-04, 07:37 PM

it is much more useful to actually point out the errors than to just say they are there. what's going on here is two differnet notions of intervals, which sort of resemble each other but not in a "rigorous" way, are being mixed up. ({}, {__}) has nothing to do with intervals like (a,b) for the first one, for one thing, is about much more than real numbers. the way i'd put it is that there is a "lattice" with {} at the bottom and {__} at the top. however, since not everything in between is "comparable," it doesn't make sense to use interval notation, in which there usually is a "total ordering," or at least a "linear ordering," involved in all elements in the interval.

His argument seems to boil down to "you can't talk about infinite sets because you can't count to infinity".
that is, i think, a straw man. that can't be his argument because he's talking about infinite sets. i also think he's referring to the absolute infinity and not just any infinite set. and in that sense, you can not count to the absolute infinity no matter what. that is to say i can prove, i think, that if P(X) is the absolute infinity, then X is the absolute infinity; hence, it cannot be achieved "from below." iow, you cannot count to it or power set to it. i think if you just unravel what organic is trying to say and get past the fact that he's using nonrigorous language, he's got kernels of truth.

in fact, he was using these statements on infinite sets like N, you know that N can be approached but never achieved. he had the right idea but what he means, i think, is the universal object, not N. when he did, i argued up until when i realized what he was really talking about.

i agree that if you change the definitions mid-sentence or mid-article, you have huge problems and that's something he has to work on but i think his nuggets of truths should be encouraged and we, like hurkyl, should be correcting the language rather than simply say it is incorrect. if that's not worth your time, i understand but if you just say it's incorrect without correction, that's not really worth organic's time.

ps: organic, they did try to correct your language but you didn't seem to listen! you have to make it clear that you're not talking about the same kind of intervals. you have to define what you mean and stick to it. look at any definition on mathworld.com or any textbook and make your definitions look like that. believe me, it's not so limiting to stick to that.

Organic

Jan8-04, 07:56 PM

Hi phoenixthoth,

Please read my previous post (my answer to Hurkyl), and reply your remarks.

Thank you,

Organic

Hurkyl

Jan8-04, 08:08 PM

Literally, "clopen" means "open and closed".

In nice situations, it turns out there aren't very many clopen sets; in some of the more common topological spaces, like \mathbb{R} or \mathbb{C}^n, the only two clopen sets are the empty set and the entire space; in nice situations the only clopen sets are those that are unions of the connected components of a space; that is, you pick a few (maybe zero) points, and you get a clopen set by taking all of the points that can be connected to the selected points.

So, for example, if I make a topological space by choosing three disjoint lines l, m, and n, then the clopen sets are \varnothing, l, m, n, l U m, l U n, m U n, l U m U n.

In not so nice situations (e.g. the rational numbers), there can be more clopen sets. For instance, the interval (\sqrt{2}, \sqrt{3}) is clopen in the rational numbers; it's closed because it contains all of its limit points in the rational numbers, and it's open because it has no boundary points in the rational numbers.

phoenixthoth

Jan8-04, 08:17 PM

and a space could be called "connected" if and only if the only clopen sets are Ø and the whole space. so when hurkyl says nice, could mean "connected."

Organic

Jan8-04, 08:22 PM

Dear Hurkyl and phoenixthoth,

First, thank you for your clear explanation about "clopen"

But what if x0 and x2 stand for infinitely many sequences where:

x0 < some given x where x is any arbitrary Rseq member.

x2 > some given x where x is any arbitrary Rseq member.

The arbitrary Rseq member represented by 01 infinitely long sequence,
taken from set Rseq, which its content = [...000,...111)XOR(...000,...111]

Rseq cunstructed by:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
Therefore it can't be "clopen", but a half closed interval.

phoenixthoth

Jan8-04, 08:49 PM

clopen means closed and open.

a half-open interval like (a,b] is neither open nor closed. this is a good example showing that sets don't fall into two categories of open or closed.

as far as
http://www.geocities.com/complementarytheory/CATpage.html
goes, i would have to say the same thing i said earlier: what is a connection in terms of something more concrete? ie, a set, a function, a category, a group, an ordered pair, etc...

Organic

Jan8-04, 09:13 PM

Please look at: http://mathworld.wolfram.com/Interval.html

My question was about this line:

"If one of the endpoints is +-oo , then the interval still contains all of its limit points, so (-oo,b] and [a,oo) are also closed intervals".

Is oo or -oo can be a notation for any collection of infinitely many objects?
---------------------------------------------------------------------------

Why AL and CR ( http://www.geocities.com/complementarytheory/CATheory.pdf ) are not concrete?

phoenixthoth

Jan8-04, 09:24 PM

no, it's not.

and oo is not an endpoint of the interval; the interval F=[a,oo) has no right endpoint.

one equivalent way to say it's closed is that if there is a sequence of points in F that converges, then the limit must be in F. this is something you can prove of F and so F is closed.

the interval G=(a, oo) is not closed because this is not the case:
the sequence {b_n}, where b_n = a+1/n, converges but not to a point in G.

master_coda

Jan8-04, 09:27 PM

Originally posted by Organic
"If one of the endpoints is +-oo , then the interval still contains all of its limit points, so (-oo,b] and [a,oo) are also closed intervals".

Is oo or -oo can be a notation for any collection of infinitely many objects?

+\infty and -\infty are just a shorthand notation being used. They do not represent actual objects.

Organic

Jan8-04, 09:50 PM

If so, then oo or -oo stands for the abstract idea of infinity.
Is it right?

master_coda

Jan8-04, 09:57 PM

Originally posted by Organic
If so, then oo or -oo stands for the abstract idea of infinity.
Is it right?

In general, \infty is the symbol used to represent some idea of infinity. However when it is used, it is used in a very rigorously defined way (if it's being used properly).

For example in [a,+\infty) it is being used as a shorthand for \lbrace x\in\mathbb{R}\colon a\leq x\rbrace. Not "the interval between a and infinity".

Organic

Jan8-04, 10:08 PM

Ho, I see.

So how can we represent this idea?

x0 and x2 stand for infinitely many sequences where:

x0 < some given x where x is any arbitrary Rseq member.

x2 > some given x where x is any arbitrary Rseq member.

The arbitrary Rseq member represented by 01 infinitely long sequence,
taken from set Rseq, which its content = [...000,...111)XOR(...000,...111]

Rseq cunstructed by:
http://www.geocities.com/complement...iagonalView.pdf

master_coda

Jan8-04, 10:12 PM

So x0 and x2 are infinite sets of sequences?

Organic

Jan8-04, 10:17 PM

Yes, and therefore can never be completed.

master_coda

Jan8-04, 10:22 PM

Well, before you can find x0 and x2, you need to define an ordering. Given two infinite sets of sequences (call them a and b), how do you determine if a<b?

Organic

Jan8-04, 10:33 PM

a XOR b can be defined as Rseq=[a,...111)XOR(...000,b]

Rseq cunstructed by:
http://www.geocities.com/complement...iagonalView.pdf

master_coda

Jan8-04, 10:41 PM

You need to provide a clear definition. That pdf just contains your issues with the diagonal method.

How do you determine if a<b?

Organic

Jan8-04, 10:53 PM

a XOR b, dont you see?

a XOR b can be defined as Rseq=[a,...111)XOR(...000,b]

Rseq cunstructed by:
http://www.geocities.com/complement...iagonalView.pdf

master_coda

Jan8-04, 11:05 PM

Your definition doesn't work. Your "interval" [a,...111) contains two different types of objects. "...111" is a sequence while "a" is a set of sequences. How do you have an interval when you have two different types of objects?

Organic

Jan8-04, 11:10 PM

No, a XOR b are some arbitrary members of Rseq=[a,...111)XOR(...000,b]

phoenixthoth

Jan8-04, 11:10 PM

Originally posted by Organic
a XOR b, dont you see?

a XOR b can be defined as Rseq=[a,...111)XOR(...000,b]

Rseq cunstructed by:
http://www.geocities.com/complement...iagonalView.pdf

i'm just not seeing what this has to do with <.

i take it that a is defined to mean [a,...111)? what does ...111 mean because that's ambiguous: aren't ...1111 and ...0111 different versions of ...111? same question for ...000.

if a means [a,...111) then why doesn't b mean [b,...111)?

i take it that [a,...111) is some kind of interval. for that to make sense, you have to explain how a<...111 without using this definition of <. i understand that the XOR between the two intervals is the "disjoint union": the set of all things in the union but not the intersection.

Organic

Jan8-04, 11:24 PM

Let's make it simpler.

a is some arbitrary member (an infinitely long 01 sequence) of Rseq=[a,...111)

Rseq cunstructed by:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Therefore a cannot be ...111, therefore Rseq cannot be completed, and therefore no transfinite number can use Rseq as its building-block.

Therefore tranfinite univereses does not exist.

master_coda

Jan8-04, 11:37 PM

You realize that to prove that something is impossible, you can't just make a single attempt to do it, fail, and then say "therefore it's impossible".

phoenixthoth

Jan9-04, 12:26 AM

Let's make it simpler.

a is some arbitrary member (an infinitely long 01 sequence) of Rseq=[a,...111)
yeah, the left endpoint. or do you mean mapped into by some kind of natural map rather than "is some member of."

Therefore a cannot be ...111

that goes without saying. in any interval G=(a,b), if x is in G, then x is not b.

therefore Rseq cannot be completed

doesn't b complete my interval G? you may want to go through either the dedekind cut or cauchy sequence of rational numbers construction of real numbers and tell us where the flaws in those proofs are.

, and therefore no transfinite number can use Rseq as its building-block.

well, since ...111 completes the interval [a,...1), the premise above doesn't hold and so this conclusion doesn't follow.

Therefore tranfinite univereses does not exist.
this doesn't follow from the past premise even if it were true. your statement "no transfinite number can use Rseq as its building-block" doesn't mean that there is no building block for transfinite universes, just that Rseq isn't one. this is what the previous poster was saying.

in short, this is not going to be a way to define a<b better than the usual ways, i don't think. please see the dedekind cut or cauchy sequence of rational numbers constructions for good ways to define a<b. in order to be something new, this theory would have to
1. define a new ordering <* such that if x and y are real numbers than x<*y iff x<y
2. be an extension of < so that <* applies to more objects than real numbers.

the question was what is an interval and i think it has been thoroughly explained. certainly not all properties of intervals have been given, but enough have been to get one's feet wet.

edit: i think you may find my treatment of cantor's diagonal argument interesting in the file attached to page 4 of "the search for absolute infinity." in a new subsets axiom, i show how there is no contradiction obtained from cantor's diagonal argument.

Organic

Jan9-04, 09:03 AM

Rseq is actually both R and N, DON'T YOU SEE THET?

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

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

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

From the other hand Rseq is N ( = 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 |N|*|N|.

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

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

Therefore transfinite universes do not hold.

Again, Rseq is both R AND N.

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(N)| exists iff P(N)=[...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, |N| is a "never ending story", therefore words like 'all' or 'complete' cannot be related to |N|.

master_coda

Jan9-04, 11:47 AM

No. Your argument is based upon your concept of infinity, not the mathematical one. Whatever contradictions you find in your concept have no bearing on mathematics.

russ_watters

Jan9-04, 12:57 PM

Originally posted by master_coda
No. Your argument is based upon your concept of infinity, not the mathematical one. Whatever contradictions you find in your concept have no bearing on mathematics. You're a little new here - have you seen all of the threads Organic started in the general math forum? You are right, but the only thing you'll accomplish by trying to explain it is headaches from banging your head against the wall. Organic is not interested in math - only in making up his own new math as he goes along.

And so far, I haven't even seen his point - math (to me) is a tool for use in science/engineering. I haven't seen where he's said what he wants to do with his new math once he's finished inventing it.

Organic

Jan9-04, 01:05 PM

No dear master_coda,

My last post clearly shows the problems that existing in Standard Mathematics about the transfinite definition.

Your last response is too weak.

NOW, YOU HAVE TO PROVE THAT MY CLIMES DO NOT HOLD.

You have no other choice, otherwise any response that can't clearly show why my argument does not hold, is meaningless.

master_coda

Jan9-04, 01:21 PM

Originally posted by Organic
No dear master_coda,

My last post clearly shows the problems that existing in Standard Mathematics about the transfinite definition.

Your last response is too weak.

NOW, YOU HAVE TO PROVE THAT MY CLIMES DO NOT HOLD.

You have no other choice, otherwise any response that can't clearly show why my argument does not hold, is meaningless.

If you don't use standard mathematical definitions, your remarks don't have anything to do with math.

Beside, in math you never have to prove someones claims don't hold. The person making the claim has to prove it does hold. And you haven't provided anything resembling a proof...you yourself admit that you have no skill at formalizing math. What makes you think you can formalize a proof?

master_coda

Jan9-04, 01:25 PM

Originally posted by russ_watters
You're a little new here - have you seen all of the threads Organic started in the general math forum? You are right, but the only thing you'll accomplish by trying to explain it is headaches from banging your head against the wall. Organic is not interested in math - only in making up his own new math as he goes along.

And so far, I haven't even seen his point - math (to me) is a tool for use in science/engineering. I haven't seen where he's said what he wants to do with his new math once he's finished inventing it.

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

Organic

Jan9-04, 02:08 PM

HO NO russ_watters,

My doors are clearly wide opened, and i clearly show the benefits of my new points of view (which are non-Euclidean) on the standard Euclidean point of view (which is based on Boolean Logic or Fuzzy Logic).

Some examples:

1) Here we can see the complementary associations between multiplication and addition.

These complementary associations, deeply changing and enriching the Number's concept.

Also we can see that a*b and b*a are noncommutative, therefore have more interesting information then the standard commutative system.

See for yourself here (please read all of it, thank you):
http://www.geocities.com/complementarytheory/ET.pdf

2) the logic bases of the above can be found here (please read all of it including all links, thank you):
http://www.geocities.com/complementarytheory/AHA.pdf

3) My general point of view on symmetry can be found here, and there we can clearly show how our standard number system is based on some private case of broken symmetry (please read all of it including all links, thank you):
http://www.geocities.com/complementarytheory/GIF.pdf
http://www.geocities.com/complementarytheory/RealModel.pdf
http://www.geocities.com/complementarytheory/LIM.pdf
http://www.geocities.com/complementarytheory/MathLimits.pdf
http://www.geocities.com/complementarytheory/SPI.pdf

4) This non-Euclidean point of view, which is based on Complementary Logic, has much more power to deal with Quantum universe, because its fundamentals are based on complementarity, redundancy, uncertainty and symmetry, which are all connected in one and simple logical system.

Also because of the same associations, which are associated by Complementary Logic, this point of view can lead us to construct and deal with much more complex systems, then Euclidean point of view can do (Because of the limitations of Blooean or Fuzzy logics).

To examine this please see (read it only if you understand Complementary Logic):
http://www.geocities.com/complementarytheory/CATheory.pdf

5) Beyond the traditional "objective" attitude to Math language, this non-Euclidean can lead us to explore new frontiers that cannot be reached by standard approach, for example (read it only if you understand Complementary Logic):
http://www.geocities.com/complementarytheory/CK.pdf
http://www.geocities.com/complementarytheory/count.pdf
http://www.geocities.com/complementarytheory/Moral.pdf

After you read and understand all of it, then and only than, please reply.

Thank you,

Organic

Organic

Jan9-04, 02:24 PM

Dear master_coda,

Notations have no meaning by themselves, we give them the meaning, and they are only tools that help us to express our ideas.

What i wrote in the post that you refuse to deal with it (by hiding behind technical excuses) is a very weak response.

This post (that you refuse to answer to it) is written in a clear Mathematical way that any Mathematician can clearly understand and can response to it.

Please show me something that I wrote in this post, which is not based on standard Mathematics.

Yours,

Orgainc

phoenixthoth

Jan9-04, 02:33 PM

Rseq is actually both R and N
so if R=Rseq=N, then R=N. this can't be because not every element of R is an element of N. 1/2, for example, is an element of R but not N.

phoenixthoth

Jan9-04, 02:48 PM

N is in bijection with Q, the set of rational numbers, so it is "equal" in a sense, but not with R. but at this point, since we're not talking about intervals anymore, perhaps you should start a new thread or continue under "combinations."

the question which was somewhat generous is how do you define a<b? i say generous because a and b were i think just defined to be "infinite sequences." i'm wondering how you can have an infinite sequence without being able to talk about infinity. but, that taken for granted, the question was asked how to define a<b. in the cauchy construction of the real numbers, real numbers a and b are equivalence classes of rational cauchy sequences. there a<b is defined.

i think you wanted to say that your ideas were like [0,oo) but they're not. oo is actually essentially undefined, though it seems like you don't define it either. that's not automatically a problem. it's just a symbol used for convienience. we could use the symbol * and it wouldn't be construed as any infinity. if you look at the definition of what it means for a limit of a sequence to converge, in the limit symbol there is an infinity but in the definition there is no mention whatsoever of infinity; therefore it can remain undefined.

real analysis is not really the proper setting (pun intended) for infinity; it is cantor's set theory and the alephs.

but if this will no longer be about intervals, perhaps you should start a new thread or continue under "combinations" or some other thread you already started. i remember asking you to show where in cantor's proof it is wrong to show that P(N)>N and you never did. in my absolute infinity theory, partly inspired by you, i did show where it would go wrong: it would fail to be a contradiction using the extension of the subsets axiom in ternary logic. if you want to see how to correctly invent a crackpot theory, read my crackpot article. just notice the overall presentation starting with definitions. you for example, defined a double-simultaneous connection as a connection such that... but you never defined connection. you have to break every definition down into something already defined or you're building a whole new undefined concept; so far, the set is virtually the only undefined concept so to add a new undefined concept would take a lot of convincing in the sense that the theory would have to have a lot of merit and power. what can you do with the theory? if it's just to disprove cantor's diagonal argument or show that transfinite sets can't exist, then not only is that incorrect, it won't get off the ground. the only way those theories can be "wrong" is if you change the axioms. but this can only be done in a way that extends them, not the other direction. my modified subsets axioms, i believe, extends the usual subsets axiom.

you've asked us to show the flaws in your claims:
1. no definitions of key terms
2. lack of rigor.

if you claim that there is a bijection from N to R, then you have to specifiy what it is or else demonstrate it exists.

Organic

Jan9-04, 04:00 PM

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

Jan9-04, 04:22 PM

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

Jan9-04, 05:12 PM

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

Jan9-04, 05:27 PM

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

Jan9-04, 06:52 PM

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

Jan9-04, 06:59 PM

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

Jan9-04, 09:33 PM

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

Jan9-04, 09:55 PM

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

Jan9-04, 10:17 PM

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

Jan9-04, 10:36 PM

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

Jan9-04, 10:41 PM

What is undefined?

phoenixthoth

Jan9-04, 10:53 PM

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

Organic

Jan9-04, 11:00 PM

Dear phoenixthoth,

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

Thank you.

Yours,

Orgainc

phoenixthoth

Jan9-04, 11:01 PM

information, for one thing.

Organic

Jan9-04, 11:06 PM

Why do I have to define it?

phoenixthoth

Jan9-04, 11:20 PM

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

Jan10-04, 11:05 AM

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

Jan10-04, 12:06 PM

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

Jan10-04, 01:16 PM

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

Jan10-04, 04:06 PM

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

Jan10-04, 04:21 PM

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

Jan10-04, 04:35 PM

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

Jan10-04, 04:46 PM

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

Jan10-04, 04:57 PM

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

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

phoenixthoth

Jan10-04, 05:21 PM

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

Jan10-04, 05:39 PM

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

Jan10-04, 05:50 PM

A definition for information:

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

phoenixthoth

Jan10-04, 06:30 PM

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

Jan10-04, 06:51 PM

"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

Jan10-04, 07:08 PM

...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.

SASs can be found here: http://207.70.190.98/toe.pdf

Organic

Jan10-04, 07:21 PM

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

Jan10-04, 07:25 PM

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

Jan10-04, 07:43 PM

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

Organic

Jan11-04, 04:38 AM

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

Jan11-04, 05:15 AM

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

Jan11-04, 12:07 PM

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

Jan13-04, 12:43 PM

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

Jan13-04, 04:37 PM

Dear matt,

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

Please check it, thank you.

matt grime

Jan14-04, 07:27 AM

Originally posted by Organic
Dear matt,

You take some of my posts in the begining of this thread, but you first have to read what happend 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 infinte 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

Jan14-04, 08:27 AM

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

Jan14-04, 09:59 AM

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

Jan14-04, 10:26 AM

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

Jan14-04, 10:35 AM

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

Jan14-04, 06:47 PM

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

Jan14-04, 06:58 PM

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

Jan14-04, 07:06 PM

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

Jan14-04, 07:17 PM

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

Organic

Jan14-04, 07:25 PM

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

Jan14-04, 07:36 PM

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

Jan15-04, 03:56 AM

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.

master_coda

Jan15-04, 09:09 AM

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

This is your definition of the word. But this is not a mathematical definition of the word.

We can ignore all definitions except relevant mathematical ones. And the one you provided is most definitely not a relevant one.

master_coda

Jan15-04, 09:14 AM

Originally posted by Organic
Before you attempt to explore the odds, be aware to the odds within you.

This isn't a very helpful quote...in order to be aware of the odds within oneself, you would have to explore the odds. Thus it would be impossible to follow this advice.

phoenixthoth

Jan15-04, 09:16 AM

the infinite set [0,1] has two "ends":0 and 1.

consider the sets x0:=Ø
and for n>0,
xn=xn-1∪{xn-1}.

a set y is considered finite if it can be put into 1-1 correspondance with an xn for some n∈N. otherwise, it is infinite.

matt grime

Jan15-04, 09:40 AM

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

This is very much abusing the idea of the axiom of infinity, which is i think equivalent to the existence of an inductive set. And it is also wrong.

You cannot induct on n to deduce things about aleph-0. This requires transfinite induction when we put ordinals in rather than cardinals. Which stats that for every successor ordinal... etc. You don't demonstrate the the veracity of the statement for all n implies it for the first infinite ordinal. And it isn't even clear what you are hoping to prove inductively.

Example:

let X be the 2 element set {0,1}

Take X(n) defined inductively by X(n) = X(n-1) \coproduct X, and X(1) = X

each X(n) is a finite set.

The limit, which i can define as the obvious filtered direct limit we will BY ABUSE OF NOTATION call X(aleph-0) is not finite, but by the axiom of induction as you want to use it, it must be! Just like you I am assigning a non-sensical meaning to aleph-0, unlike you I both define the induction and how to take the limit.

Go through your proof again, it is incorrect. It is seemingly the basis for your decision to develop your complementary logic - this boolean logic cna't deal with infinity stuff.

Why does it bother you that there is no largest number, that a list of the naturals will not terminate?

You say that one can not apply the word all to an infinite set. Let N be the set of Natural numbers. IN what way is it not complete? You can't just give a 'but it's not' answer, you must demonstrate that your assertions are meaningful by backing it up with evidence, or a proof or a definition. This is not philosophy.

Organic

Jan15-04, 10:28 AM

Matt,

You wrote:

This requires transfinite induction

Transfinite induction does not exist because if you force the system beyond its ability to be described by infinitely many objects, then you have no mathmatical tools that can deal with the actual infinity, which is the content of {___}.

1-1 map, or any other mathematical tool can work only among collections of finitely or infinitely many objects.

I'll be glad if you show me how you can use math tools and get an input, when you have {__} content as your information source.

As much as I see it no mathematical tool can deal with the content of {___}.

Therefore no meaningful input can be found and used beyond the potential infinity (a collection of infinitely many objects, which their fundamental property is not to include their end).

Aleph0 can be used only as a cardinal of N objects, where |N| value obeys the lows of probability, as I clearly demonstrate here:

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

master_coda

Jan15-04, 10:47 AM

Originally posted by Organic
(a collection of infinitely many objects, which their fundamental property is not to include their end)

As I have already said, this is not true. You've even been given an example showing that this is not true.

Organic

Jan15-04, 10:51 AM

master_coda,

I am talking about "far" (objective) and "close" (subjective) odds.

Without knowing the "close" one and its influence on you as an explorer, you can't deal with the "far".

Organic

Jan15-04, 10:57 AM

master_code,

no base_value > 1 is out of the lows of probebility.

Pelease look again in my example here:

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

Organic

Jan15-04, 11:15 AM

Hi Dear phoenixthoth,

Please give you comments on my last post to Matt Grime.

Thank you.

matt grime

Jan16-04, 06:23 AM

Originally posted by Organic
Matt,

You wrote:

Transfinite induction does not exist because if you force the system beyond its ability to be described by infinitely many objects, then you have no mathmatical tools that can deal with the actual infinity, which is the content of {___}.
[\QUOTE]

I'm afraid you are slightly missing the point here. I don't care whether transfinite induction is used or not. You are the one who wants to induct on things in terms of infinite ordinals, though you keep using cardinals

Let's ignore transfinite induction as it is irrelevant

[QUOTE]
1-1 map, or any other mathematical tool can work only among collections of finitely or infinitely many objects.

[\QUOTE]

As all collections of objects are either infinite or finite in size, what are you attempting to exclude here?

[QUOTE]
I'll be glad if you show me how you can use math tools and get an input, when you have {__} as your information source.
[\QUOTE]

I'd be glad if you explained what you mean by this phrase. Hint define {__}. Is it any infinite set? How about N that's an infinite set. Here's a function on it (you might even get a kick out of this one)

card(N) = aleph-0

There you go, how's that for a function? It's defined on the set of cardinalities, which is infinite (though you might disagree I have no prob;em with that) and it's input is an infinite set too. As it is also unclear whether you want the domain to an infinite set or the domain to be a set of infinite sets I thought I'd put this doubly useful example in.

[QUOTE]
As much as I see it no mathematical tool can deal with the content of {___}.

Therefore no meaningful input can be found and used beyond the potential infinity (a collection of infinitely many objects, which their fundamental property is not to include their end).

Aleph0 can be used only as a cardinal of N objects, where |N| value obeys the lows of probability, as I clearly demonstrate here:

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

What if the set doesn't have an end? That's what the infinity in [0, oo) means in the examples you've been misusing. There doesn't have to be limit.

Organic

Jan16-04, 08:40 AM

Dear Matt,

My answer to there http://www.physicsforums.com/showthread.php?s=&postid=130493#post130493 is also good here.

matt grime

Jan16-04, 09:17 AM

And that answers whcih question?

thanks to me buggering up the quotes, there were lots in there you might have missed. let me reiterate.

[QUOTE]
1-1 map, or any other mathematical tool can work only among collections of finitely or infinitely many objects.

[\QUOTE]

As all collections of objects are either infinite or finite in size, what are you attempting to exclude here?

So we now know that your infinity is the north pole of Riemann's Sphere?

Ok, you want some functions on the Riemann sphere whcih can cope with this infinity?

how about the set of Mobius transformations?

z maps to (az+b)(cz+d)^{-1}

there is a very well understood way of dealing with infinity in there: it maps to a/c, and the point -d/c is sent to infinity.

Now, then this infinity is not the infinity of the axiom of infinity, which doesn't actually have a concrete 'infinity' in it, merely tells you when a set is infinite.

The concept of infinite set and the point at infinity are not the same thing. You should check you understand the difference between the adjective infinite and the noun infinity. That could explain a lot.

Let's reiterate: the point at infinity is not the infinity used when talking about sizes and cardinalities. If you wish it to be so in your system then something is going to need a lot of explaining.

Perhaps we are now seeing where your confusion lies.

Organic

Jan16-04, 09:41 AM

Matt,

A point in infinity is ONE.

matt grime

Jan16-04, 09:53 AM

So putting it all together

the point at infinity (of the Riemann sphere) is ONE, which is your name for 'the full set' opposite to the empty set? That is, one point, the one point that compactifies the complex plane is an infinite set, infact _the_ infinite set that is complementary to the empty set? Next you'll be saying the empty set is 0. Oh, you have haven't you, now I come to think of it! It the empty set was at the bottom of those cells after repeatedly dividing in half.

I don't know if this is sensible, but how does it relate to the infinity of an infinite set like the naturl numbers?

Organic

Jan16-04, 02:56 PM

Matt,

Please read this:

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

But be aware that through my point of view, aleph0 is the cardinal of Z*, N or Q , and its value is under the lows of probability.

Now please look again on my symmetrical point of view:

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

Thank you.

Orgainc

matt grime

Jan16-04, 05:39 PM

And that answers the question how?

Aleph-0 is not under any laws of probabilty, in a non-trivial way. It is well defined.

Organic

Jan16-04, 06:58 PM

{___} content which is ONE infinitely long object, is unreachable by any collection of infinitely many objects, and it is the top limit of Math language.

Its oppsite is the "content" of {} which is the bot. limit of Math language, and it is unreachable by any content of a non-empty set.

Shortly speaking: ({},{__}):={x|{}<--x(={.}) AND x(={._.})-->{__}}.

Please read again this including all its links:

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

Aleph0 value is under the lows of propability, because no collection of infinitely many objects can reach the contents of {} or {__}.

matt grime

Jan16-04, 07:14 PM

Just give one example of a set which is the same as {__} just one, that's all.

then explain what it means to reach these limits you are so fond of, please. Go on just for little old me define 'reach'. I know a picture's worth a thousand words, but each post is up to 10,000 words which is ten pictures.

Organic

Jan16-04, 07:30 PM

Define example.

matt grime

Jan16-04, 07:34 PM

Originally posted by Organic
Define example.

Seeing as you like models, give one set in your model of your thery which obeys the rules you have for {__}.
If you prefer something that is a realization. If no such exists you have a vacuous theory.

Is R a set which is ONE or one of many? by your dichotomy theory it is one of these.

Organic

Jan16-04, 07:49 PM

The rule is symmetry.

If {} then {__}.

matt grime

Jan17-04, 06:17 AM

Is that giving an example? Did I not explain sufficiently what example means?

That propostion above does not make sense.
First rule of writing mathematics: all sentences should be sentences.

We appear to have established that {__}, actual infinity, ONE, is the North Pole of the Riemann Sphere. Is this correct? The last of these isn't a set. Does that matter? It is also an infinitely long object, but I think this is you using long in some undefined way. How can a point have length?

I think you need to differentiate in your mind between the empty set and zero, and infinity as a point lying further away from zero than all real (complex) numbers and infinite meaning not finite. You appear to be comparing two different concepts.

In that vein, try explaining how your infinity {__} is related to cardinality. How does it relate to, say, infinite dimensional vector spaces.

Now why is it important to find an injection between this point at infinity and say R?

I think you should start a new thread to go through this from the beginning because it is getting too complicated with all these different topics. So start one about the New Diagonal Argument and we'll sort that out shall we?

Organic

Jan17-04, 07:13 AM

Dear Matt,

Thank you for your advice.

Your model is about point, but not mine.

If you read my papers (and you don't) for example:

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

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

You will find my very simple definition to continuum (or fullness) which is clearly separated from any point model.

Also if you read this:

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

Where I clearly and simply distinguish between ONE(={__})
and one of many(={.}) .

matt grime

Jan17-04, 12:00 PM

I have read your articles. They are not very clear in what you are trying to express.

You have said that {__} is the point at infinity on the Riemann Sphere, or that it represents it in some posting mentioning RiemannsBall.pdf

the cardinality of {__} in your universe is 1, for instance, it is also decribed as an infinitely long line.

Try putting some more words in your articles so that you explain how the deductions you've made have been arrived at, as they currently read like a bunch of unmotivated random assertions from someone who can't understand maths properly.

Organic

Jan17-04, 01:55 PM

Again, there is no objective Math that you understand it or not, all what we have is a rigorous agreement based on language.

You cannot understand my work because you think 2D, and my system is fractalic multi-D and Non-Euclidean.

One of its first results is the difference between multiplication and addition, and their complement associations, which are some of the fundamentals of my number system.

If you did not see it until know, and you are totally closed in your 2d point of view, which is the Euclidean-Mathematics point of view.

matt grime

Jan17-04, 02:14 PM

That's a rather strange attitude to take. I don't understand your math because of its paucity of intellectual content and inability to be self-consistent or explicable. One of the things you've said is that it isn't necessary to be rigorous, and you've repeatedly demonstrated that by refusing to answer questions and explain what's going on, write your ideas out clearly in this forum, define anything from first principles, pay even the slightest attention to conventions and already extant notions/notations.

In fact I can't think of one question you've ever answered to the satisfaction on the questioner.

Here are some outstanding ones:

what does it mean for a set to be too strong to be measured?

explain the inductive process you use to create the list of 01 combinations and prove that it is complete (except for ...111)

is {__} a set or a proper class? what are some of its elements?

explain what you mean by 1-1 map because it is apparent that you believe there are no 1-1 maps from an infinite set to itself that are also surjections.

define opposite

what is the mathematical meaning of fullness?

how can an inequality be equal the the empty set, or a set containing as an element an inequality (whether it is valid or not) be equal to the empty set?

in what way is aleph-0 probabilistic?

take as many words to explain these as possible. don't post pdfs. As you are posting in a public forum one presumes you are attempting to explain to people what you believe (and that you think it worthwhile sharing), so it seems ony fair to explain everything properly; how can people be expected to understand it otherwise.