Closed Intervals with Infinite Endpoints: Explained

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The discussion centers on the nature of closed intervals in real analysis, particularly those with infinite endpoints, such as (-∞, b] and [a, ∞). Participants argue that these intervals are closed because they contain all their limit points, as any convergent sequence within these intervals converges to a point also within the interval. There is a debate about the definitions of closed sets and intervals, with some arguing that the inclusion of infinite endpoints complicates this classification. The conversation also touches on philosophical implications of infinity in mathematics and the limitations of traditional mathematical logic. Ultimately, the discussion highlights the complexities of defining closed intervals and the broader implications of infinity in mathematical theory.
  • #91
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?
 
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  • #92
  • #93
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.
 
  • #94
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.
 
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  • #95
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.
 
  • #97
The fact that your constuction fails just means that your construction was bad. It doesn't mean that other constructions must also fail.
 
  • #98
Dont you see that by using ZF axiom of infinity on the power value of 2^x, x=aleph0 by standard math notation?
 
  • #99
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.
 
  • #100
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.
 
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  • #101
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.
 
  • #102
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.
 
  • #103
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.
 
  • #104
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.
 
  • #105
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
 
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  • #106
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.
 
  • #107
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".
 
  • #109
Hi Dear phoenixthoth,

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

Thank you.
 
  • #110
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

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?

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.


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.
 
  • #112
And that answers whcih question?

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

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.
 
  • #113
Matt,

A point in infinity is ONE.
 
  • #114
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?
 
  • #116
And that answers the question how?

Aleph-0 is not under any laws of probabilty, in a non-trivial way. It is well defined.
 
  • #117
{___} 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 {__}.
 
  • #118
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.
 
  • #119
Define example.
 
  • #120
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.
 

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