Re: Cantor Confusion


Virgil schrieb:


If real numbers are to be represented by sequences that any two
sequences which are "arbitrarily close" represent the same number.

Therefore Cantor's proof is invalid. With increasing length of the
list, the difference introduced by exchanging the diagonal becomes
smaller and smaller. For an infinite list it vanishes at all.

For example, as when one represents the same number in different bases.

And, representatives are
*not* limits. When considering the equivalence classes, most sequences
have one limit: the equivalence class it is sitting in.

I think that you are still thinking that *some* represenation defines
a real number; but that is not the case.

In Cantor's list there are those unique representations required.

Not so. Even in decimal, Cantor's diagonal rule allows for certain
rationals having dual representation.

Which two numbers could that be?


Therefore I do not understand why you say "Numbers are fixed entities".
They are merely defined by sequences.

The sequence 1 + 1/2 + 1/4 + 1/8 + ...+ 1/2^n + ... "defines" a fixed
number. That that number has other representations does not make that
number into a variable quantity.

The sequence 1 - 1/3 + 1/5 - 1/7 +-... does not define a fixed number.

Regards, WM

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