Re: The real numbers, and general comments

From: Dave Seaman (dseaman_at_no.such.host)
Date: 10/10/04 

Date: Sun, 10 Oct 2004 14:33:02 +0000 (UTC)

On 9 Oct 2004 23:46:41 -0700, Andrew Usher wrote:
> Dave Seaman <dseaman@no.such.host> wrote in message news:<ck62lq$kjn$1@mozo.cc.purdue.edu>...

>> >> I meant that the power set axiom is an axiom of ZF. That is what the theorem
>> >> is based on.

>> > No, it is _based on_ logic; it may be proved within ZF.

>> It can't be proved without an axiom guaranteeing the existence of a power
>> set.

> False, the diagonalisation proof implies no such thing. It, rather,
> directly constructs a real number not enumerable.

I was talking about Cantor's theorem. That's the theorem that says |X| <
|P(X)| for every X. You can't even make the statement in the first place
unless you know that P(X) exists.

But if you want to talk about the uncountability of the reals, then you
should know that the construction of the reals (whether by Dedekind cuts
or by Cauchy sequences) also depends on the power set axiom. It's a
construction of ZF.

>> Pretty much everything in mathematics is based on ZF or on ZFC. That
>> doesn't mean the axioms of ZFC make any reference to groups, or vector
>> spaces, or topological spaces, or Riemann surfaces. They also don't
>> mention models.

> This is a ridiculous use of 'based on'. Of course, you can force
> proofs to depend on a certain 'model of logic', but that does not
> imply that the concepts do.

ZF is not a model of logic. ZF is a set of axioms for set theory. There
are other ways of establishing the foundations (category theory, for
example), but ZF is almost universally accepted for that purpose.

>> > Could you please state exactly what it does say? (I.e. does it mean
>> > only 'there exist countable models)

>> It has been explained elsewhere in the thread. The point you are missing
>> is that a countable model can contain sets that are uncountable according
>> to the model.

> It has not been completely explained. Your last point is precisely
> what I deny is meaningful.

In any model of ZF, and for any X in that model, we can say that |X| <
|P(X)| in the model. In particular, this means that P(N) is uncountable
in that model, even if the model itself is countable. That means there
is no bijection *in the model* between N and P(N) *in that model*, even
though both may be countable as seen from the outside. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

Relevant Pages

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