Closed Intervals with Infinite Endpoints: Explained

[Organic]

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

My question is 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".

How come ?

 

[master_coda]

If a sequence within [a,\infty) converges it must converge to a value within [a,\infty), and so the interval is by definition closed.


Suppose you have a sequence that converges to p. If p<a, then the sequence must contain a point in the interval (p,a) and so the sequence is not actually in the interval.

So if the sequence is in the interval we must have p\geq a. Thus p\in[a,\infty). Thus any convergent sequence in the interval converges to a point in the interval. Thus the interval is closed.

 

[NateTG]

It depends on what definition of closed you're using, but (-\infty,b] is indeed a closed set in \Re under the usual topology.

If you think of closed as 'contains all of its own limit points', then you can see that
(-\infty,b]
does indeed contain all real limit points of sequences of reals that only contain numbers from (-\infty,b].

This is relatively easy to prove:
Let's say we have a sequence S such that s \in S \rightarrow s \in (-\infty,b], and that x is a limit point of S.
Assume, by contradiction, that x is not in (-\infty,b]. Then clearly x > b. Now, because x is a limit point of S, for any \epsilon > 0 there exists some s' \in S with |s'-x| < \epsilon, but if \epsilon=\frac{x-b}{2} then there cannot be any suitable s' \in (-\infty,b]

Alternatively, if you start with the notion that (a,b) is an open set in \Re then you can see that (b,\infty) is an open set, since it is \Cup_{x \in \Re, x>=b} (x,x+1). Then it's compliment (\infty,b] must be closed.

P.S. Sorry, I don't have the tex for union handy.

 

[HallsofIvy]

If you are thinking that (-\infty,b] and [a,\infty) do not include all limit points because they do not include -\infty or \infty, remember that the those are not in standard real number system. That is why we never say "[-\inft,b] or [a,\infty].

"[a,\infty)" really means "a and all real numbers larger than a".

 

[HallsofIvy]

If you are thinking that (-\infty,b] and [a,\infty) do not include all limit points because they do not include -\infty or \infty, remember that the those are not in the standard real number system. That is why we never say "[-\infty,b] or [a,\infty].

"[a,\infty)" really means "a and all real numbers larger than a".

 

[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

 

[phoenixthoth]

i wouldn't say that oo or -oo in real analysis refers to infinitely many objects, at least no more than do numbers like 1 or square root of 2. actually, in the cauchy sequence construction and dedikind cut construction of R, all numbers are sets with infinitely many objects but of the same kind of infinity. neither oo nor -oo are real numbers, they aren't really defined except that oo is a symbol one could interpret as infinity if one wants such that x<oo for all x in R.

quote from my real analysis book:

the symbols +oo and -oo are used here purely for convienience in notation and are not to be considered as being real numbers.

the convienience is, for example, when talking about intervals like
G={x in R : x>a}. now if F={x in R : a<x<b}, then the notation is (a,b). for G, this notation would become awkward: (a, (period)
so to make the two notations look alike, we just say (a, oo) when we actually mean (a, (period) oo is not the right endpoint of G; there is no right endpoint of G.

 

[Organic]

Hi

My argument is very simple:

1) Things get infinitesimally small and never reach {}.

2) Things get infinitely big and never reach {__}.

Therefore things are in ({},{__}).

{__} is the full set, which its content is an infinitely long non-factorized-one line.

{__} is the opposite of {}, and vice versa.


Please see again this example:

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

Therefore no infinitesimally or infinitely many elements can be related to words like all or complete.

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

The idea of transfinite universes is meaningless.

 

[master_coda]

Except that none of the things you're talking about have anything to do with math.

 

[Organic]

No dear master_coda,

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 goes beyond the above article.

Please try to understand this:

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

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

 

[phoenixthoth]

1) Things get infinitesimally small and never reach {}.

2) Things get infinitely big and never reach {__}.

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?

 

[NateTG]

Well, the simple response is that the 'things' which Organic refers to are not mathematical objects.
It follows that notions about big and small things are also not mathematical and so on.

Hence, the argument that Organic makes is not mathematical in nature.

For mathematical associations to be drawn -- for example the implied notion that there is a correlation between Euclidean (ie. Plane) geometry and boolean logic, and Non-Euclidean geometry and non-boolean logic -- they need to be formalized, or at least described.

 

[master_coda]

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 explanation 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]

(For some reason I couldn't post this morning, but here's what I was going to 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]

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]

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.../ClopenSubset.html+math+clopen&hl=en&ie=UTF-8

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

 

[phoenixthoth]

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]

Hi phoenixthoth,

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

Thank you,

Organic

 

[Hurkyl]

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]

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]

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]

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]

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]

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]

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]

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

 

[master_coda]

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]

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]

So x0 and x2 are infinite sets of sequences?

 

[Organic]

Yes, and therefore can never be completed.

 

[master_coda]

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&lt;b?

 

[Organic]

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]

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

How do you determine if a&lt;b?

 

[Organic]

a XOR b, don't 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]

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]

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

 

[phoenixthoth]

Originally posted by Organic
a XOR b, don't 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.