Re: Review of Mueckenheims book.

On 28 Feb., 14:47, Carsten Schultz <cars...@xxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx schrieb:





On 26 Feb., 03:27, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1172390012.159715.55...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> > > Why can't you repeat that definition of a well-ordering?

> > Because it depends on the model.

> I am asking for a well-ordering of the reals, i.e., *the real numbers*
> as we all know them from mathematics, not for some elements of an
> artificial model. And I am stating that this well-ordering does not
> exist.

In that case you have to precisely define what you *mean* with "the real
numbers as we all know them". I would think of them as objects definable
from ZF.

The real numbers should are well known. They have been defined from
ZF. But our mathenmatics is not a model of ZF.

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:
> Let us choose s to be the set of all well-orderings of the real
> numbers; by AC s is nonempty. Yet one cannot prove in ZFC that s
> contains any definabel member as one cannot prove in ZFC that there is
> a definable well-ordering of the set of all real numbers. Fraenkel,
> Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations of Set
> Theory", 2nd edn., North Holland, Amsterdam (1984), p. 69.

And if you mean something else, then state it explicitly please and
remove your assertion of an error of mine.
And, so what? You stated that "it has been proven that a definable
well-ordering of the reals does not exist". Now you state something
different: "it can not be proven that a definable well-ordering does
exist". The two are *not* the same.

So what?
If it is proven that one cannot prove it, then it cannot be done.
Because doing it would be a proof that it can be done.

While you shy away from mathematically defined terms to make your claim
sound more plausible, it is still wrong.

A definable set is a set given by a condition P(x) on x without
parameters and such that in ZF, or in ZFC, one can prove that there
exists just a single element x which satisfies this condition. It is
just in this case that we can speak of "the set x such that P(x)
holds" and give it a proper name. For example, the null-set O is given
by the condition "x is memberless" and the set Z* is given by the
condition "x is a subset of every set Z which contains O and which for
each of its members y also contains {y}".
Let us choose s to be the set of all well-orderings of the real
numbers. (because of AC it is nonempty) yet one cannot prove in ZFC
that s contains any definable member as one cannot prove in ZFC that
there is a definable well-ordering of the set of all real numbers.
If there exists exactly one x which fulfils P(x) then this x is not a
well-ordering of R.

If we can prove of an ordering that it is a definable well ordering
within the framework of the ZFC axiom alone, then the above statement
is wrong. Or better: then there is a contradiction in ZFC.

While your lack of understanding ist mostly sad, your audacity can be
mildly entertaining.

Regards, WM

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