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From: Bill Taylor on 28 Jun 2010 01:52
On Jun 21, 1:08 am, Rupert <rupertmccal...(a)yahoo.com> wrote:

> There is a minimal well-founded model for ZF.
> It is equal to L_alpha for a countable ordinal alpha.

So what (else) is known about this particular alpha?
How does it compare with other countable ordinals?

-- Wondering William

** Starting at zero,
** Accumulate where needed:
** Ordinal heaven!
From: Bill Taylor on 28 Jun 2010 02:00
On Jun 21, 2:30 am, George Greene <gree...(a)email.unc.edu> wrote:

> The important difference between ZF and PA here is that in the case of PA,
> the standard model is the minimal one, whereas in the case of ZF, what
> people generally intuit as standard, as the INTENDED domain of discourse,
> is a MAXIMAL model.

So I understand. But I have often wondered why.

>There sort of "obviously are" more sets than constructible ones.

My feeling would be that there were "obviously fewer".
Indeed, only those up to L(w_1^CK). But I seem to be a lone voice
here.

> Non-standard naturals (by contrast) are so deprecated that people will
> blithely allege that it is possible for theories such as PA+~G
> to have "false" theorems.

Well I guess they implictly mean, "false in the intended model",.
which is so, isn't it? And, i would say, a reasonable usage of
"true".

-- Befuddled Bill

** Mathematics runs under set theory
** just as Freecell runs under windows.
From: Daryl McCullough on 28 Jun 2010 12:31
Bill Taylor says...
>
>On Jun 21, 2:30 am, George Greene <gree...(a)email.unc.edu> wrote:
>
>> The important difference between ZF and PA here is that in the case of PA,
>> the standard model is the minimal one, whereas in the case of ZF, what
>> people generally intuit as standard, as the INTENDED domain of discourse,
>> is a MAXIMAL model.
>
>So I understand. But I have often wondered why.

Well, if you have a minimal model, then you can perform various
operations to get a new model that has more elements than the
minimal model, and, unlike the case with the nonstandard naturals,
these new elements have just as much right to be called "sets" as
any others.

--
Daryl McCullough
Ithaca, NY

From: George Greene on 28 Jun 2010 12:39
On Jun 28, 2:00 am, Bill Taylor <w.tay...(a)math.canterbury.ac.nz>
wrote:
> On Jun 21, 2:30 am, George Greene <gree...(a)email.unc.edu> wrote:
> >There sort of "obviously are" more sets than constructible ones.
>
> My feeling would be that there were "obviously fewer".

Please. You don't get a feeling here. You can just diagonalize, or
use forcing.
Other bigger models clearly MUST exist, otherwise the Axiom of Choice
(since it is true in the inner model) WOULD be PROVABLE!

> Indeed, only those up to L(w_1^CK).

WHOA, HOSS.
That is WAY too many. You can add an axiom to ZFC insisting that
there is a non-
constructible set and still have A COUNTABLE MODEL of it, at first-
order, anyway.
The issue isn't "height" (L(whatever-transfinite-ordinal-you-choose))
-- it's BREADTH.

> > Non-standard naturals (by contrast) are so deprecated that people will
> > blithely allege that it is possible for theories such as  PA+~G
> > to have "false" theorems.
>
> Well I guess they implictly mean, "false in the intended model",.
> which is so, isn't it?

Yes, it's so, but it's a lot less reasonable than it would be in ZFC.
PA is a small denumerable KIND of thing. ZFC by contrast is huge,
even with V=L.

>  And, i would say, a reasonable usage of "true".

For a countable first-order theory, it is almost not even reasonable
TO HAVE an
"intended model". If you intend fewer of the models than exist, then
THE REASONABLE
and rational thing to do IS ADD AN AXIOM to RESTRICT attention to the
model(s)
you DO intend! The main reason this hasn't been done in PA is that G2
proves that
it's a wild goose chase (you will in some sense Never be able to pin
it down to
"just the natural" numbers). But once you decide not to add an axiom,
ALL
the models of the axioms you DO use are in some sense "created equal".
Saying "true" and "false" (as opposed to provable or disprovable) is
just plain
discriminating against non-standard models. You don't GET to claim
that they
were unintended. The current ocean-floor oil spill in the Gulf of
Mexico wasn't
intended either, but that doesn't mean it's not happening.

>
> -- Befuddled Bill
>
> **   Mathematics runs under set theory
> **   just as Freecell runs under windows.

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