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From: Frode Bjørdal on 12 May 2010 17:41
On 12 Mai, 23:04, Frode Bjørdal <fbenlightenment4...(a)gmail.com> wrote:
> On 11 Mai, 15:37, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> > Rupert <rupertmccal...(a)yahoo.com> writes:
> > > Suppose we consider only the standard models, and your question
> > > becomes: What is the smallest countable ordinal which can be aleph-one
> > > in the sense of some countable transitive model of ZFC? I do not know
> > > whether any special name has been given to this ordinal. It is larger
> > > than the smallest non-recursive ordinal, obviously.
>
> > I don't think anyone's found it in their heart to bestow a special name
> > on the ordinal in question, the aleph-1 of the minimal model. I spent a
> > minute trying to come up with something interesting to say about it,
> > but, alas, failed. Perhaps it has some interesting closure properties?
> > Absent an ordinal analysis of impredicative set theory we shall in all
> > likelihood remain forever in the dark.
>
> Unless I misunderstand something here, this would, in case it exists,
> be the
> smallest countable ordinal d  so that L(d) is a model of ZF.Perhaps it
> should
> have been named the Cohen ordinal? If we follow the scarequoted
> terminology
> of the Wikipedia entry on large countable ordinals, this ordinal would
> be an
> "unprovable" ordinal..

On second thought: Perhaps the question is for the smalles ordinal e
so that the minimal
model proves that e is aleph-1? I believe, as for my previous
interpretation, or misinterpretation,
that one, as suggested by Aatu Koskensilta, will remain in the dark.
This suggests that there perhaps
is no standard model for ZF.
From: George Greene on 18 May 2010 00:31
On May 5, 4:08 pm, Marc Alcobé García <malc...(a)gmail.com> wrote:
> ZFC is a first-order theory with a countable model. This means there
> is an interpretation of the axioms for which the universe of discourse
> is a countable set. I wonder if any of this questions makes any sense:
>
> 1. How can that model contain all ordinals?

It doesn't.

> 2. How can that model contain all countable ordinals? (there are
> uncountably many of them)

Exactly.


> If the explanation of 2 is that simply the bijection between aleph_1
> and omega does not exist:

It's not SIMPLY that.
It's that that bijection does not exist IN THIS MODEL.
It still exists. Or rather, there still exists a bijection between
THIS MODEL'S version of aleph_1 and omega. But, again,
that bijection does not exist IN [the domain of] THIS MODEL.


> 3. Which is the least countable ordinal that can play that role?

What role?? Your antecedents are unclear.
You were asking about the case where a certain bijection did not
exist,
so "that role" would seem to mean the role of that bijection.
NO countable ordinal is EVER going to play THAT role; ordinals are not
bijections.


From: George Greene on 18 May 2010 00:34
On May 11, 9:37 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> I don't think anyone's found it in their heart to bestow a special name
> on the ordinal in question, the aleph-1 of the minimal model.

Is "the minimal model [for ZFC]" even well-defined??
Is it the inner/constructible model?
From: Daryl McCullough on 18 May 2010 06:20
George Greene says...
>
>On May 11, 9:37=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>> I don't think anyone's found it in their heart to bestow a special name
>> on the ordinal in question, the aleph-1 of the minimal model.
>
>Is "the minimal model [for ZFC]" even well-defined??
>Is it the inner/constructible model?

Yes. Godel's description of the constructible universe gives a
minimal interpretation of the power set operation (it only includes
*definable* subsets). If ZFC is consistent, then iterating this
definable power set operation will produce a model of ZFC after
some countable number of iterations. So there is some kappa such
that L_kappa is a model of ZFC, and no subset of L_kappa is a
model of ZFC.

--
Daryl McCullough
Ithaca, NY

From: Aatu Koskensilta on 18 May 2010 09:07
stevendaryl3016(a)yahoo.com (Daryl McCullough) writes:

> Godel's description of the constructible universe gives a minimal
> interpretation of the power set operation (it only includes
> *definable* subsets). If ZFC is consistent, then iterating this
> definable power set operation will produce a model of ZFC after some
> countable number of iterations.

Consistency is insufficient; we need the existence of a standard model.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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