Re: Countable models of ZFC
- From: Rupert <rupertmccallum@xxxxxxxxx>
- Date: Fri, 05 Oct 2007 23:14:21 -0700
On Oct 4, 8:52 pm, george <gree...@xxxxxxxxxx> wrote:
On Oct 3, 8:35 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
Of course every domain of a model is a set. That is not a problem.
It is also not universally AGREED, OBVIOUSLY.
OBVIOUSLY, from ITS OWN viewpoint, the domain
of EVERY model of ZFC IS A PROPER CLASS AND NOT
a set. I already posted here an excerpt FROM A BOOK BY
WELL-RESPECTED AUTHORS in which THEY wanted to
reserve the right to make the domain of a model (ESPECIALLY,
one would think, the domain of a model OF A SET OR CLASS
THEORY) a proper class. Precisely as THEY noted, that certainly
causes notational difficulties, if you are going to make that proper
class suddenly contained IN some structure (not SUPPOSED to do
THAT with proper classes) whose OTHER contents are interpretations
of predicates and functions OVER that class/domain.
Yes, sure, it is possible to have a framework where some models are
proper classes. I don't like working in such a framework myself, and I
wasn't. If you do work in such a framework then you can talk about the
"intended model" of set theory. But there's still no need to mention
it in this context. The definition was fine as it was, and it was
clear, it was not a "glib gloss".
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.
You can't just SAY that something is wrong!
You HAVE to explain why.
You snipped the rest of my response. If it is inadequate at engaging
with what you said, then *you* have to engage with it and explain why.
You can't just snip it.
In any case, there are at least 3 parts to what I just said, and
one might hope you could concede the legitimacy of at least one of
them:
If M is a set THEN THERE IS an outer model.
Sigh. Well, in the part which you snipped I said "For me models are
sets". You apparently want to talk about the intended model, which is
a proper class. All right, if we allow proper classes there is an
intended model, and we can call it the outer model if you want. But
what's the point of talking about it? What does it add to the
discussion? The definition was fine by itself.
THERE IS a model of
set theory IN WHICH M is a set. There IS a DOMAIN of THAT model
of which M is an element. THAT model has a membership relation
already.
And when YOU talked about restricting some membership relation to M,
it was THAT membership relation that you were restricting.
AS I HAVE ALWAYS SAID, THERE ARE *2* epsilons in the scenario
that YOU presented! And *I* am *NOT* the one who was CONFUSED
about this point!
I don't think I ever denied that from one point of view it might be
reasonable to say that there are two epsilons. I said it was trivial
and irrelevant. What is the point of bringing it up?
.
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- Re: Countable models of ZFC
- From: george
- Re: Countable models of ZFC
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- Re: Countable models of ZFC
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