Closed Intervals with Infinite Endpoints: Explained

[master_coda]

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]

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]

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

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

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

 

[matt grime]

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]

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]

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]

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]

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]

Hi Dear phoenixthoth,

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

Thank you.

 

[matt grime]

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.

 

[Organic]

Dear Matt,

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

 

[matt grime]

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.

 

[Organic]

Matt,

A point in infinity is ONE.

 

[matt grime]

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]

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]

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]

{___} 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]

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]

Define example.

 

[matt grime]

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]

The rule is symmetry.

If {} then {__}.

 

[matt grime]

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]

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]

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]

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]

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.

 

[Organic]

Dear Matt,

Your world is (0 XOR 1).

My world is fading transition between (0 XOR 1) and (0 AND 1).

Your world is a private case of my word.

I cannot translate my definitions to your world for example:

Because your logical world is limited to 2D (0 XOR 1) and my world is not limited to 2D logic, when you ask me to define my system in terms of your logical 2D word, I hope that you understand that when it is translated, her point of view is lost.

So, instead of continuing these useless replies between us, I am going to open a new thread, and the I'll ask the members to show us what is the difference between multiplication and addition by using Boolean logic.

 

[master_coda]

Organic, I would suggest that you are a megalomaniac, but I'm afraid that you'll just make up your own definition of what "megalomaniac" means and it'll become a complement.

Perhaps your world would be more meaningful if it tooks concepts from our world such as "consistency" and "coherence". Those things have done wonders in our world.


Perhaps if you start connecting to the Internet with a computer built on principles of Complimentary logic instead of Boolean logic, your credibility will increase. Building such a device should be a simple task for one with an intellect such as yours.

 

[Organic]

Dear master_coda,

Maybe you have some example to this:

https://www.physicsforums.com/showthread.php?s=&threadid=12783