Re: Review of Mueckenheims book.

mueckenh@xxxxxxxxxxxxxxxxx schrieb:
On 3 Mrz., 03:47, Carsten Schultz <cars...@xxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx schrieb:
Would a definable well-ordering of the reals be found, using no
further axioms than ZFC, then it could be proved in ZFC that a
definable well ordering exists. Can you agree so far?
Then it was proven in ZFC that such a well-ordering exists. Then above
statement was wrong. Can you follow so far?
Do you understand the difference between a statement that can not be
proven in ZFC and a statement whose negation can be proven in ZFC?

That can be a difference. I case of a definable well ordering there is
no difference. If it exists, then it is the proof of its existence.
Then the statement: "a definable well ordering can be proved" is
correct, and the statement "a definable well ordering cannot be
proved" is wrong.

Let me get this straight. We are talking about a fixed formula P such
that in ZFC "there is exactly on x with P(x)" can be proved. The claim
is that

P(x) => x is not a well-ordering of R

cannot be proved in ZFC. Do you claim that it would follow that

P(x) => x is a well-ordering of R

can be proved in ZFC? If not, what is your claim?


If it exists, but has not yet been found, there is no difference.

Whatever that means.

Perhaps you will also be able to say us how it can be proven, *using
nothing but the axioms of ZFC*, that x is a well-ordering of R?
What makes you think I had claimed such a thing? Please see also a
couple of lines below.
What is a well-ordering which cannot be proven to be a well-ordering?
This line is nonsense.

Why can't you simply admit that the statement in your book should have
been formulated more carefully?

Because my book is correct in this point, if Fraenkel et al. are
correct.

Do you at least see by now that your book could be read to have claimed
something at this point that Fraenkel et al. did not?

Best regards,

Carsten

--
Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
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