Re: Ultimate debunking of Cantor's Theory

On Jul 12, 8:44 am, Calvin <cri...@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.

No, this doesn't work. In fact, it fails spectacularly:

a(1) = .011111...
a(2) = .011111...
a(3) = .011111...
....

Your "new" binary decimal turns out to be .100000..., which is equal
to .01111...; so you don't get a new number after all!

Are there other noteworthy forms of the diagonal argument?

The Halting problem?

....

This is as good of a place to any to present my motivation for this
thread: (1) I had read somewhere about the "Infinity Axiom" and the
fact that you can have a set theory without (actual) infinite sets,
and I was wondering if it actually works; and (2) I was sick and tired
of anti-Cantor threads in sci.math.

--- Christopher Heckman

.

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 ... A related theorem and proof is the theorem that for any set S, ...
    (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: What does "Infinity" really mean?
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  • Re: basic first-order model theory question.
    ... >> Saying that the reals are denumerable is certainly wrong. ... Cantor's diagonalization shows that some infinite sets _aren't_ ... The set of all positive integers are denumerable. ... > That was my reasoning from the previous post. ...
    (sci.logic)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... > Infinite sets are not equivalent. ... we expect from set theory that a well-ordering of the ... > we do not know how to spell out an order to put the reals in that is ... properties of the previous and next points in the total well-ordering ...
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