How can we extend the concept of sets to deal with infinite dimensions?

[Organic]

No theoretical system can survive without being aware to its limitations.

It means that any x output can be only a model(X) input.

Shortly speaking, x=model(X).

Math is first of all a form of theory, therefore any concept that can be used by it is only a model(CONCEPT).

For example, let us take infinity concept.

If INF is infinity itself (= actual infinity) , then inf=model(INF)=potential infinity.

Please look at this model for better understanding:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

In this way we first of all aware to our input limitations, which are:

No input = model(EMPTINESS) = lowest limit.

No input = model(FULLNESS) = highest limit.

If we translate this to set's representation then:

{} content = model(EMPTINESS) = lowest limit.

{__} content = model(FULLNESS) = highest limit.

Between these limits ({},{__}) we can find inf=model(INF)=potential infinity, where inf has two input forms:

{.} = singleton, which is a localized element.

{.__.} = non-singleton, which is a non-localized element (connect at least two different singletons).

{.} and {._.} can appear in two basic collections:

Collection {a, b, c} is finitely many elements.

Collection {a, b, c, ...} is infinitely many elements (=inf) .

Any non-empty collection which is not a singleton, is an association between {.} and {._.}, for example:

              b   b
             {a , a}    
              .   .  
              |   | 
              |___|_
              |    
                
           
             {a , b}    
              .   .  
              |   | 
              |___|
              |

For more details please look at:

http://www.geocities.com/complementarytheory/CATpage.html


I'll be glad to get your remarks and insights.


Thank you.


Organic

 

[matt grime]

'infinity itself'

hmm. would you care to explain what this means?

or for that matter explain what infinity the concept means to you?

 

[Organic]

In this case actual infinity can be no input.

 

[matt grime]

No, please state clearly what it is that you think the concept of infinity is that you to have a model for.

 

[Organic]

Please try to understand this, any explanation by some language is only the model of the explained thing.

For example: no explanation of simplicity is simplicity itself, and also no explanation of infinity is infinity itself.

Therefore no theory can deal with the thing itself but only with the explanation of the thing.

An explanation of a thing is what I call a model of a thing.

The best way I have found to express this idea, is not by words but by:

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

I'll be glad if you find a way to translate this picture model to model of words.

 

[matt grime]

That is the point at infinity, the one point compacification of the complex plane (well, real line).

It is NOT 'the infinity' in some abstract sense. it is "the point at infinity". this is the standard thing that people fail to grasp. by abuse of notation it is often labelled infinity, but it is not the infinity that people mean in the sum from one to infinity, it is not the infinity invoked when one says there are an infinity of possible solutions. it is not the infinity used when one speaks of 'the edges of the universe', it is not the infinity of (-infinity,0] - look at the definition of that, realize infinity is not used in the inequality defining it, but that the infinity is a useful short hand.



Infinity is not a well defined mathematical entity. infinite is what one ought to use, but using the term infinity is a convenience.

It means different things at different times, if every one would use the word infinite properly then we wouldn't need to have these problems.

 

[Organic]

You are talking about what I call a rigorous agreement between people.

I the first page of my website I wrote:

Dear researcher,

Math, in my opinion, is first of all a rigorous agreement that based on language.

Symmetry is maybe the best tool that can be used to measure simplicity, where simplicity is the best platform for stable agreement.

Any agreement must be aware to the fact that no model of simplicity is simplicity itself.

This awareness to the difference between x-model and x-itself is the first condition for any stable agreement, because it gives it the ability to be changed.


As for your question about what is 'infinity itself' my best answer at this moment is:

'infinity itself' is some example of a non-approachable thing (or the limit) of any theoretical system.

 

[Organic]

http://mathworld.wolfram.com/Infinity.html

http://mathworld.wolfram.com/Infinite.html

As you can see the word 'quantity' is used, so Math forces these concepts to be under its wings.

 

[matt grime]

Perhaps you should stop taking wolfram as gospel.

That is one use for a symbol labelled infinity. In fact it is just defining the symbol $$\infinity$$'s use in real analysis, that the ratio 1/x grows without bound as x tends to 0+.

 

[MathematicalPhysicist]

so the model of x is actually a function of it?

 

[Organic]

Please look again at:

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

The infinitely long base of the empty triangles is a representation of actual emptiness.

The infinitely long base of the green triangles is a representation of actual fullness.

Math cannot use them as an input.


Another example:

A meaningful circle with radius r exists between r=0 and r=oo.

When r=0 or r=oo, we have no circle.

Now instead of a 'circle' we can write 'some finite or infinite collection of elements'.

{} is equivalent to r=0 and {__} is equivalent to r=oo.

 

[Loren Booda]

To relate scientifically, one must translate feelings and images of beauty into rigorously applied mathematics and operable physics, which are beautiful in themselves. That evolution may take many years, and many revisions of an adaptive model.

 

[Organic]

Hi Loren Booda

You wrote:

To relate scientifically, one must translate feelings and images of beauty into rigorously applied mathematics and operable physics, which are beautiful in themselves.

I think you know that "one good picture = 1000 words"

But more to the point, let us examine a little part from your work.

You write:

Inflection point: p''[x]=0

6x+2b=0

x=-b/3

As we can see, you use notations 6,3 and 2 in your equations.

Please let me ask: what 6, 3 or 2 you use?

And I am asking this because there are 2 structural possabilities of 2, 3 structural possabilities of 3 and 76 structural possabilities of 6.

Please see for yourself:

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

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

 

[matt grime]

Originally posted by Organic Please look again at:

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

The infinitely long base of the empty triangles is a representation of actual emptiness.

The infinitely long base of the green triangles is a representation of actual fullness.

Math cannot use them as an input.

would math want to?

Another example:

A meaningful circle with radius r exists between r=0 and r=oo.

When r=0 or r=oo, we have no circle.

Now instead of a 'circle' we can write 'some finite or infinite collection of elements'.

we *can*, but how on Earth can you justify what follows is an equivalent statement, or infact meaningful in any sense? When did we become able to replace words at will in a sentence, and claim the new sentence is EQUIVALENT. I might perhaps accept analogous. But that's not the same, or am I just reading too much mathematical meanin into equivalence?

{} is equivalent to r=0 and {__} is equivalent to r=oo. [/B]

Well, you theory asserts that current maths language is unable to handle sets because it doesn't allow for probability. As probability is defined using current maths (and sets) don't you think that statement a little mendacious?

 

[Organic]

Dear Matt,

would math want to?

Do you know what math wants?

we *can*, but how on Earth can you justify what follows is an equivalent statement, or infact meaningful in any sense? When did we become able to replace words at will in a sentence, and claim the new sentence is EQUIVALENT. I might perhaps accept analogous. But that's not the same, or am I just reading too much mathematical meanin into equivalence?

Please take this part of my previous post only as an analogy.

Well, you theory asserts that current maths language is unable to handle sets because it doesn't allow for probability. As probability is defined using current maths (and sets) don't you think that statement a little mendacious?

Current Math language using probability, but not as a fundamental property of Set's and Number's concepts.

Through my point of view, this is one of the most important insights that have to be transferred from modern physics to pure Math language.

 

[matt grime]

but th probabiltiy you cite is defined in terms of sets, to require sets to be defined probabilistically is circular.

 

[Organic]

It is not, because my new information structures are at least complementary associations between {.} AND {.__.} fundamental structures.

Please show me this point of view in standard sets.

 

[matt grime]

in order to use probability in the foundations of your set theory, you must give a set theoretic independent way to define probability, which currently you can't because probabilty is defined as by measure on a probability space; they all require sets in their definition.

 

[Organic]

I use symmetry-degree to define probability, again you are looking at my work from your point of view about probability.

Semmetry-degree clearly represented by the information-sturctures, which based on associations among {.} and {._.} fundamental elements.

Again please look at:

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

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

Please show me a conventional way to costruct my system by using the current set's definitions.

 

[matt grime]

of course I'm using standard interpretations of the words 'probability' and 'set'. as you aren't perhaps you should consider using a different name ofr these different objects?

how can i construct something in my model which is 'fundamentally' different from yours when the thing you ask me to construct is your 'fundamentally' different object? that isn't an issue, like i keep saying, you are free to develop whatever theory you want, just don't misuse the old one as you repeatedly do. so pick different names to avoid confusion - that is common sense.

 

[Organic]

But what you call a set is a private case of 0 redundancy and 0 uncertainty information structure of my system, therefore my system is a natural extension of the current set's concept.

So, why should I give my information structure a different name?

For example Let us say that Gset is a framework to deal with geometrical objects.

Let us say that current geometry is only 0, 1 and 2 dimensions and we find a way to show that there are infinitely many dimensions, which are not just integers but fractional by nature too.

In this case I don’t see any reason not to continue using Gset’s framework.