Bruno Marchal skrev:

Le 15-nov.-07, à 14:45, Torgny Tholerus a écrit :

Do you have the big-black-cloud interpretation of "..."?  By that I mean that there is a big black cloud at the end of the visible part of universe,

Concerning what I am trying to convey, this is problematic. The word "universe" is problematic. The word "visible" is also problematic.

and the sequence of numbers is disappearing into the cloud, so that you can only see the numbers before the cloud, but  you can not see what happens at the end of the sequence, because it is hidden by the cloud.

I don't think that math is about seeing. I have never seen a number. It is a category mistake. I can interpret sometimes some symbol as refering to number, but that's all.

A way to prove the consistency of a theory is to make a "visualization" of the theory.  If you can visualize all that happens in the theory, then you know the theory is consistent.

To visualize the natural numbers, you can think of them as a long sequence {0,1,2,3,4,5,...}, and this sequence is going far, far, away.

But you can only visualize finite sequences.  So you can think that you have a finite sequence of numbers, and you have a big black cloud far, far, away.  You see the first part of the sequence {0,1,2,...,m} before the cloud.  But inside the cloud you can imagine that you have the finite sequence {m+1,m+2,...,4*m-1,4*m}.  This whole sequence {0,1,2,...,m,m+1,...4*m} is what you call the set N of all natural numbers.

>From that set N you construct the true subset {0,2,4,6,...,2*m,2*m+2,...,4*m}, which you call the set E of all even numbers.  The visible part of the set E is then {0,2,4,...,2*m}, and the hidden part of that sequence is {2*m+2,...,4*m}.

Now you define a new concept INNFINITE, that is defined by:

If you have a bijection from all visible numbers of a set S, to all visible numbers of a true subset of S, then you say that the set S in INNFINITE.

Then you can use this concept INNFINITE, and you will get a consistent theory with no contradictions, because you have a finite visualization of this theory.


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