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From: Bill Taylor on 20 Jun 2010 02:30
As it says, this may be a stupid query. Be gentle.

1st-order PA has many models. But it seems to me that there is
a very clear "minimal model", in that the standard model is
(isomorphic to) a subset of any other model. And I gather
that this fact can be proved with a fairly trivial extension
of PA itself, extended into extremely basic model theory
(i.e. set theory).

Assuming I am not yet too haywire:- Does this notion also
apply to ZF? Is there in some sense "a minimal model"?
Is this easy to prove? As easy as for PA?

I'm pretty sure I've read at some time a reference to
a "minimal model" for ZF, probably in Cohen '66.
But I don't know if this is the same thing.

TIA.

-- Baffled Bill
From: Rupert on 20 Jun 2010 09:08
On Jun 20, 4:30 pm, Bill Taylor <w.tay...(a)math.canterbury.ac.nz>
wrote:
> As it says, this may be a stupid query.  Be gentle.
>
> 1st-order PA has many models.  But it seems to me that there is
> a very clear "minimal model", in that the standard model is
> (isomorphic to) a subset of any other model.  And I gather
> that this fact can be proved with a fairly trivial extension
> of PA itself, extended into extremely basic model theory
> (i.e. set theory).
>
> Assuming I am not yet too haywire:-  Does this notion also
> apply to ZF?   Is there in some sense "a minimal model"?
> Is this easy to prove?  As easy as for PA?
>
> I'm pretty sure I've read at some time a reference to
> a "minimal model" for ZF, probably in Cohen '66.
> But I don't know if this is the same thing.
>
> TIA.
>
> -- Baffled Bill

There is a minimal well-founded model for ZF. It is equal to L_alpha
for a countable ordinal alpha. This is indeed discussed in Cohen '66.
From: George Greene on 20 Jun 2010 10:30
On Jun 20, 2:30 am, Bill Taylor <w.tay...(a)math.canterbury.ac.nz>
wrote:
> As it says, this may be a stupid query.  Be gentle.
>
> 1st-order PA has many models.  But it seems to me that there is
> a very clear "minimal model", in that the standard model is
> (isomorphic to) a subset of any other model.  And I gather
> that this fact can be proved with a fairly trivial extension
> of PA itself, extended into extremely basic model theory
> (i.e. set theory).
>
> Assuming I am not yet too haywire:-  Does this notion also
> apply to ZF?   Is there in some sense "a minimal model"?

Yes. The minimal model is called the "constructible sets".
The wikipedia article is
http://en.wikipedia.org/wiki/Constructible_universe

> Is this easy to prove?  As easy as for PA?

As the article explains, it was proved by Godel in 1938.
I don't know when (or by whom, or with how much machinery)
it was proved that all models of PA have to contain N as a submodel.
Maybe you will trade me a link for that.


> I'm pretty sure I've read at some time a reference to
> a "minimal model" for ZF, probably in Cohen '66.
> But I don't know if this is the same thing.

Godel '38 and Cohen '66 are bookends of the proof of the independence
of the Axiom of Choice
from ZF. Godel '38 proves that Choice is true in the inner model
(which proves that Choice
is Not Disprovable from ZF). Cohen '66 forces the existence of a
model of ZF in which
Choice is false (which proves that Choice is Not Provable from ZF).

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


From: Frederick Williams on 20 Jun 2010 10:42
Bill Taylor wrote:
>
> [...] Is there in some sense "a minimal model"?
> Is this easy to prove?

See Shepherdson, 'Inner Models for Set Theory I, II, III', JSL, 1951,
1952, 1953.

If there is a standard model of ZF which is a set S, then the class of
constructible sets in S is a minimal (Shepherdson says 'inner') model.

(When Shepherdson was my tutor I felt obliged to read his publications.
I don't remember much about 'Inner Models', but I remember that.
(Unless, of course, I've misremembered.))
--
I can't go on, I'll go on.
From: Rupert on 21 Jun 2010 06:07
On Jun 21, 12:30 am, George Greene <gree...(a)email.unc.edu> wrote:
> On Jun 20, 2:30 am, Bill Taylor <w.tay...(a)math.canterbury.ac.nz>
> wrote:
>
> > As it says, this may be a stupid query.  Be gentle.
>
> > 1st-order PA has many models.  But it seems to me that there is
> > a very clear "minimal model", in that the standard model is
> > (isomorphic to) a subset of any other model.  And I gather
> > that this fact can be proved with a fairly trivial extension
> > of PA itself, extended into extremely basic model theory
> > (i.e. set theory).
>
> > Assuming I am not yet too haywire:-  Does this notion also
> > apply to ZF?   Is there in some sense "a minimal model"?
>
> Yes.  The minimal model is called the "constructible sets".
> The wikipedia article ishttp://en.wikipedia.org/wiki/Constructible_universe
>

The constructible universe is a proper class model. It is the smallest
transitive class model of ZF which contains all the ordinals. I think
that what Bill Taylor had in mind was the smallest transitive set
which is a model of ZF. This is a fragment of the constructible
universe.

> > Is this easy to prove?  As easy as for PA?
>
> As the article explains, it was proved by Godel in 1938.
> I don't know when (or by whom, or with how much machinery)
> it was proved that all models of PA have to contain N as a submodel.
> Maybe you will trade me a link for that.
>
> > I'm pretty sure I've read at some time a reference to
> > a "minimal model" for ZF, probably in Cohen '66.
> > But I don't know if this is the same thing.
>
> Godel '38 and Cohen '66 are bookends of the proof of the independence
> of the Axiom of Choice
> from ZF.  Godel '38 proves that Choice is true in the inner model
> (which proves that Choice
> is Not Disprovable from ZF).  Cohen '66 forces the existence of a
> model of ZF in which
> Choice is false (which proves that Choice is Not Provable from ZF).
>
> 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.  There sort of "obviously are" more sets than
> constructible ones.
> 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.

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