Re: Ultimate debunking of Cantor's Theory

In article <1184247886.869334.260540@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Calvin <crice5@xxxxxxxxxxxxxx> wrote:

On Jul 11, 11:24 pm, "Peter Webb"
<webbfam...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
But set theory minus the Axiom of infinity is a perfectly valid set theory.
The advantage for the poster is that Cantor's diagonal construction of the
Reals doesn't exist, or indeed any form of the diagonal argument applied to
infinite sets. ...

I assume that by 'Cantor's diagonal construction' you
mean considering a hypothetical countable list of all of
the decimal expansions of the real numbers between 0 and
1, then going down the diagonal, changing each digit to
any digit other than the one at each diagonal position,
and then noticing that the real number so constructed by
using the changed diagonal elements cannot be in the
original list. Thus the existence of a countable list
of decimal expansions of the reals between 0 and 1 is
disproven.

A variation of that which I subjectively like is making
it a list of binary expansions instead of decimal. Then
it is only necessary to 'flip' the diagonal, changing
all ones to zeros and all zeros to ones.

Are there other noteworthy forms of the diagonal argument?

A related theorem and proof is the theorem that for any set S, there is
no surjective function from it to its power set, P(S) (the set of all
subsets of S).

Suppose f is a function from some S to its power set P(S), f: S --> P(S)
consider the subset of S of all members which do not map into their
images under f, i.e., M_f = {s in S: not s in f(s)}.

Given any f:S -->P(S), such a set is always well defined, but a moment's
thought shows that one cannot have f(x) in M_f for any x in S.
.

Relevant Pages

  • Re: Ultimate debunking of Cantors Theory
    ... The advantage for the poster is that Cantor's diagonal construction of the ... Reals doesn't exist, or indeed any form of the diagonal argument applied to ... of decimal expansions of the reals between 0 and 1 is ... fact that you can have a set theory without infinite sets, ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... The advantage for the poster is that Cantor's diagonal construction of the ... Reals doesn't exist, or indeed any form of the diagonal argument applied to ... of decimal expansions of the reals between 0 and 1 is ... all ones to zeros and all zeros to ones. ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... The key difference between Cantor's diagonal proof with the reals ... When we diagonalize we get an infinite length string... ... Finite length decimal expansions are a subset of rational numbers, ...
    (sci.math)
  • Re: ******* TRY THESE SCI.MATH **********
    ... decimal expansions, it only seems to be confusing you. ... you seem to be mixing the issues of uncountability of the reals ... The issue of computability is worse: you can easily define a single ... digit be 1 if the n-th Turing Machine halts, ...
    (sci.math)
  • Re: ******* TRY THESE SCI.MATH **********
    ... decimal expansions, it only seems to be confusing you. ... you seem to be mixing the issues of uncountability of the reals ... The issue of computability is worse: you can easily define a single ... digit be 1 if the n-th Turing Machine halts, ...
    (sci.logic)
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