Re: Review of Mueckenheims book.

In article <1172755635.480182.187560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 1 Mrz., 03:07, "*** T. Winter" <***.Win...@xxxxxx> wrote:
....
> The real numbers should well known. They have been defined from
> ZF. But our mathematics is not a model of ZF.

You state so.

There is no model of complete ZFC at all. Hence our mathematics cannot
be a model.

In what ways are the models of ZFC incomplete? What axiom do they not
include?

> You wrote about an error on p. 88:
> Set theory is described, both in the older (Cantor time) form as in
> the modern form. However, I think there is an error on p. 88. It
> is stated that (translated): "and it is even provable that a
> well-ordering [of the reals] can not be defined at all..."
> My understanding is that a definable well-order is not inconsistent
> with ZF, and there are indeed models were a well-order can be
> defined.
>
> If your understanding concerns a definable well ordering of the reals,
> then you are wrong:

No.

So you know better than Fraenkel et al.? At least you should give that
well-ordering here.

There is no contradiction. And, as I stated again and again, the well-
ordering depends on the model. But apparently you do not know how models
work, nevertheless you use Skolem's countable model to prove that the
reals are countable. Remarkable.


> > And models are important, as
> > (whichever way you look at it) you are looking from a model.
>
> No.

Opinion? Or something else?

Our mathematics is not a model of ZFC.

Oh.

> There is no definable well ordering of the real numbers. That is
> clearly correct, accoring the Feferman's and Levy's work quotet in:
> Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel:
> "Foundations of Set Theory", 2nd edn., North Holland, Amsterdam
> (1984), p. 69.

Wrong. Your statement implies that it can be proven within ZF that a
definable well-ordering can not be given. If that were the case, that
is a theorem within ZF, and so also in every model of ZF. But that is
clearly false, because there are models of ZF that have a definable
well-ordering.

But not of the real numbers.

In that case your definition of real numbers is outside ZF. And, indeed,
the standard definition requires additional axioms that are not axioms
in ZF.

> Please remove your wrong statement immediately.

Not shown wrong, request refused.

Show a definable well-ordering of the real numbers of currect
mathematics. Here and now! Or withdraw your untenable claim.

Pray follow the links I have provided that *show* definable well-orderings
within ZF.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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