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From: George Greene on 18 May 2010 15:53
On May 18, 6:20 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> 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.

Is it known which way this model decides the continuum hypothesis?
From: George Greene on 18 May 2010 15:55
On May 18, 9:07 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> Consistency is insufficient; we need the existence of a standard model.
>

So we cannot prove that if there is a model, then there is a standard
model?
"Standard" here means well-founded and what else?
This is another one of those times where the tyro presumes-from-
ignorance
to contradict the academic culture on a purely terminological point:
the world
would be a better place if there were ONE standard model (as re PA).
Surely the model people MEAN when they speak of THE cumulative
hierarchy
is the one with [platonically] "full" powersets.
From: Daryl McCullough on 18 May 2010 16:39
George Greene says...
>
>On May 18, 6:20=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> 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.
>
>Is it known which way this model decides the continuum hypothesis?

Yes, in this model, there are no uncountable cardinals smaller than
the continuum.

--
Daryl McCullough
Ithaca, NY

From: Aatu Koskensilta on 20 May 2010 05:00
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes:

> 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..

The least alpha such that L_alpha is a model of set theory is not a
recursive ordinal.

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

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 20 May 2010 05:04
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes:

> 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.

This suggestion needs be spelled out in more detail. On the face of it
there's nothing in the aleph-one of the minimal model to recommend the
idea there's no standard model of set theory.

--
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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