Re: Countable models of ZFC



Rupert wrote:
On Oct 3, 9:34 pm, george <gree...@xxxxxxxxxx> wrote:
On Sep 26, 5:35 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:

I'm sorry you had so much trouble understanding it. It seems pretty
straightforward to me. Given a set M,

Well, WHICH are we being given?? A SET?? OR A MODEL??

the standard membership relation
on M is {(x,y):x, y in M and x in y}.

Up to this point, M is a set.

So a model (M,

And THERE is the problem.
Now, SUDDENLY, M is a model INSTEAD OF a set.
Or rather, M is the domain of a model.


Of course every domain of a model is a set. That is not a problem.

It is a problem. It is an epistomological problem.

If you are using the word "model" in a certain technical sense
proper to model theory, the the domain of a model is, by
definition, a set. No problem.

If you are using the word "model" to mean, roughly, "something
that this formal system X is true of" then a model of ZFC,
that is, something of which ZFC is true, may not be a set.
That's the problem.

ZFC is a first-order theory. A first-order theory talks
about its universe of discourse, presumably non-empty. The
universe of discourse consists of the things the quantifiers
of the theory range over.

A "model" of ZFC, in the looser, non-model-theoretic sense, is
any thing or things that the theorems of ZFC are true of. These
will be the things that quantifiers range over. These things that
are quantified over are conventionally referred to as "sets".

But is the universe of discourse a "set"?

We know that if ZFC is consistent, if it has any models at all,
then there are different things that it is true of: there are
distinct models of ZFC. It may be that the universe of discourse
may be a "set" in some of these models, i.e., that the domain
of discourse of one model may be one of the things the quantifiers
range over in another model.

But it has not been shown, and seems doubtful, that the domain
of every model, in this looser sense of "model", is an element
of some other model -- although in a different, and technical
sense of the word, the domain of every model is, by definition,
a set.

Please correct me if I'm wrong.


If M is both a set AND the domain of a model
then IN ADDITION to the model (M,whatever),
THERE MUST EXIST ALSO
the model OF SOME SET THEORY in which M is a set,
and that model must have a domain of which M is a member.
THAT MODEL ALSO has a membership relation,
and THAT MODEL ALSO might be standard OR NONstandard.

No. This is wrong.

For me, models are sets.

But what is a set? A set is what the axioms are true of.
That's all.

You no doubt have an idea about what _you_ mean by set. ZFC
has _its_ idea of what a set is -- or rather, ideas. If
ZFC is coherent, there are _different things_ that answer to
ZFC's description of what a set is.

I don't like talking about proper classes.
When I say "M is a set", I don't mean it's a set living in some model,
I just mean it's a set, period.

Some people might want to use a theory which allows proper classes and
then the universe counts as a model too. Of course, this model is
standard (just as the standard kilogram weighs one kilogram).

You have to be able to make sense of "x is a member of y" all by
itself,

It is the axioms that makes sense of "x e y". Unfortunately, perhaps,
they leave room for different interpretations.

There is no doubt that the intended interpretation is what we
ordinarily mean in natural language by the words "is a member of".
It is assumed that the axioms capture this intended meaning,
among others.

not just "x is a member of y in such-and-such a model". There
has to be such a notion available, or your semantics will never get
off the ground.

IF THAT model was nonstandard then merely restricting its membership
relation to some submodel IS NOT going to guarantee that that submodel
(in this case, (M,E)) is standard. SOME nonstandard models DO have
nonstandard submodels.

If you're going to take the view that you can never understand any
sentence unless you've specified what model it's relativized to,
you're going to tie yourself in knots.

The truth of some sentences of the language of ZFC are relative to
a model. There's no getting around that.

Oh, hell, I guess I'll go ahead and hit the "send" key.

--
hz
.

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