Re: Review of Mueckenheims book.

On Feb 21, 2:02 am, mueck...@xxxxxxxxxxxxxxxxx wrote:
On 20 Feb., 20:09, "MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:

On Feb 20, 3:44 am, mueck...@xxxxxxxxxxxxxxxxx wrote:

Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
had a proof that set theory is wrong.

Unless you give a mathematical definition of 'wrong', I don't know
what you mean by 'a proof that set theory is wrong'.

Burali-Forti showed that there is no greatest ordinal. In Z set
theories, it is not a contradiction that there is no greatest ordinal.

Russell showed that Frege's system is inconsistent. That does not
impinge on Z set theories.

Skolem showed that a theory can have a countable model and also a
theorem that there are sets that are uncountable. But it is not a
contradiction that a theory has a countable model and also a theorem
that there are sets that are uncountable.

In any model of ZFC the ZFC axioms are valid.

Some writers do use the word 'valid' in the sense of 'true'. But it
helps to keep the distinction. In any model of ZFC, the ZFC axioms are
true. A formula is valid iff it is true in every model for the
language.

With them there exists
the empty set (usually by axiom).

With what? The models are not objects of the theory but rather of the
meta-theory (well, there are inner models too, but that's a bit more
(or less?) complicated). Also, I already told you that with the schema
of separation as as an axiom schema or theorem schema, we don't need
an empty set axiom.

With the empty set there exists the
set omega (by axiom f infinity) and with omega there exists the power
set of omega (by axiom of power set). Why should there exist all these
elements but not a mapping, not even the identity mapping?

Are you talking about Skolem's paradox now? First, what identity
mapping are you referring to? There always exists a bijection of a set
onto itself. As to other mappings, they exist or they don't as
provided by the axioms of the theory. And we have to be careful to be
specific as to what requirements we place as to in WHICH sets a
mapping does or does not exist. If you (generic 'you') just blur such
important distinctions, then you get crank claims of contradictions
that are not contradictions.

Tarkski and Banach showed that ZFC has a certain result that is
puzzling at first glance. But it requires the axiom of choice; it is
not a contradiction; and (as far as I understand) would only
contradict a physical theory if matter had infinite density.

Put a marble in your living room and pray that ZFC may be valid
tonight. Perhaps your prayer is heard and our universe is full of
marbles tomorrow. Then I will believe in ZFC. Otherwise you should
begin to suspect it.

ZFC is not a theory of physics. I don't know much about the Tarski-
Banach theorem, but I have not read that it entails anything contrary
to physics. As I understand, the theorem does not render that a
PHYSICAL object may be perform in the way the ABSTRACT object in the
theorem performs.

Also, I notice that you did not respond to my other points in reply to
you. And I notice that is a habit you have of brushing off from among
the more obviously solid points of rebuttal to you. Of course, I
understand that no one can be expected to answer every line of every
post. But I do note that I've written quite a bit to you, with a lot
of tissue of explanation, yet you demur from following up on those
substantive matters as, what seems to me, to be a way of avoiding an
engagement of ideas that would lead you into considerations beyond
your program of denunciations of set theory. My point is not to go
back over all those posts, but rather to suggest that we discuss these
points for all they are worth and not just as fodder for your
continued harrumping that set theory is "wrong".

Of course this all would not affect any physical theory. --- And as
infinite density of matter cannot happen in physics it would be
nonsense to have it in a physical theory other than an idealization.

I was told by another poster that my use of 'density' is not correct
here. Nevertheless, I see that we agree on this point (at least to the
extent that it is a given that matter does not have whatever property
it is allows the Tarski-Banach behavior).

The point you keep missing is that set theory does NEED to be intended
as a theory in which we can read off each of its sentences as a
statement about physical objects.

MoeBlee

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