The power of the transfinite system

[Hurkyl]

One additional comment; if a new system includes an old system inside it, then all of the results of the old system must also be results of the new system.

 

[Organic]

I like your attitude about being careful when we try to develop a new logical system.

Therefore my basic attitude is to find ways to associate between opposite concepts in such a way that at least they do not contradict each other.

And after that we can check if they can associate and define more interesting results then the state of not being associated.

This is the main idea of Complementary Logic.

Because I am not familiar with the standard formal mathematical notations form one hand, and I did not find any existing model in pure Mathematics from the other hand, I had no choice but to write my ideas in the best way I can, which is not an easy task for professionals to understand it, and I am aware of it.

One of the things that I cared about was to use the simplest possible way to organize my ideas.

For example, I associated in a coherent way between concepts like redundancy, uncertainty and symmetry to construct a very simple model of symmetry break levels.

Then I have found that addition and multiplication are complement operations, and there is a beautiful and simple way to order them when using their complement associations on each other.

My non-formal paper with some examples can be found here
(Hyrkyl helped me to write the first 9 lines of it):

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

I am here in this forum to share my ideas with you, and learn from your experience, remarks and insights.

I think no man's work can really be done alone, and maybe one of the most beautiful and meaningful (and also powerful, therefore dangerous) languages, which is Math language, has to be developed by team work.

 

[matt grime]

Originally posted by Organic
master_coda,

I am not talking about the second (f: A --> A) I am talking about the meaning of being a collection of infinitely many objects.

So, when A is a collection of infinitely many objects,
its identity map (f: A --> A) = (f: A --> B) , where B is a proper subset of A.

But this is exactly what I clime about the paradox which appears contrary to expectations, and if you read this http://www.geocities.com/complementarytheory/Identity.pdf
I am sure that you will understand my argument.

No, organic, you are wrong, that is not what is meant at any of those Wolfram links you posted. You evidently don't understand the maths here.

 

[Organic]

Dear Matt,

If you say that I don't understand the math here, then first you have to show that you understand my point of view, and only than you can show what is wrong in this point of view and how we can correct it.

Please correct me.

 

[matt grime]

I'm not talking about your theory, I'm talking about the correctness of your view on Cantor's criterion for being an infinite set.

Ok, you say that a set is infinite if

the identity map Id:A ---> A

is EQUAL to a bijective map f:A ---> B

for B a PROPER a subset.

Now, maps are surjective onto their image, the image sets are not EQUAL, therefore the maps cannot be EQUAL. If you dispute this then you are not using the correct definition of EQUAL.

Alternative proof. If f: A --> B where B is a proper subset of A, then there is some x in A not mapped to x, otherwise the image is not a proper subset. however, the definition of the identity map is that Id(x) = x for all x, so the maps are not equal.


This is not using your theory, this is to do with you not understanding mathematics as almost everyone else does. I'm not touching on your point of view in the slightest.

 

[Organic]

Matt what you wrote is clear and beautiful.

Can you show some interesting results that are based on the difference between these non-equal maps?

Thank you.

 

[matt grime]

Yes, how about Rickards criterion for Epaisse subcategories:

If T is a traingulated category, and S a ful triangualated subcategory closed under arbitrary coproducts, then it is closed under taking summands (akin to the eilenberg maclane swindle)
+ denotes direct sum

Let X be in S if X = Y + Z in T, form the infinite direct sum

Y+Z+Y+Z+Y+Z... call this A. A is in S by construction and is isomorphic to the infinite direct sum of X with itself, which is clearly isomorphic to

Z+Y+Z+Y+Z... call this B.

there is then a natural map from A to B by shifting components one to the right

as S is closed under triangles, the third corner must be in S, but this is just Z. Hence X=Y+Z in S too.

the shift map is the equivalent of the bijection to a proper subset.

If you want a baby version, just take the left and right shift operators on L^{infinity}

right then left shift is the identity, left then right isn't, so there is a map with a left inverse which is not a right inverse and vice versa.

 

[Organic]

Please give another intersting result that using this difference between left and right shifts, thank you.

Edit 1:

I have an idea about this non-symmetric shift.

When we deal with the identity map we don't care about some possible difference that can be between each pair included in the map.

Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .

Where can I find some paper that deals with what I call pairs_possible_difference?

Edit 2:

I have another idea based on the difference between

pairs_possible_difference = 0
XOR
pairs_possible_difference > 0

Multiplication and Addition are the same only when pairs_possible_difference = 0.

What do you think?

 

[matt grime]

Originally posted by Organic
Please give another intersting result that using this difference between left and right shifts, thank you.

Edit 1:

I have an idea about this non-symmetric shift.

When we deal with the identity map we don't care about some possible difference that can be between each pair included in the map.

Beg your pardon, this is nonsense again.

Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .

what does it matter if its ordered or not? the set might not even be ordered

Where can I find some paper that deals with what I call pairs_possible_difference?

I have no idea because its something you just invented, and I haven't got a clue what you mean by it

Edit 2:

I have another idea based on the difference between

Oh bugger, I've not given you more things to ruin, have I?

pairs_possible_difference = 0
XOR
pairs_possible_difference > 0

why do you have this obsession with XOR all the time. why don't you use more words?

Multiplication and Addition are the same only when pairs_possible_difference = 0.

What do you think?

What I think isn't fit for a public forum right now.

 

[Hurkyl]

Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .

Are you talking about the elements of A that are not in B?

More precisely:

Let A be a set.
Let f be a 1-1 map from A into itself.
Define B to be the range of f, so that f is a bijection from A to B, and B is a subset of A.

Are you trying to talk about the elements that are in A but are not in B?

 

[Organic]

In both cases A has the elements of B.