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From: Aatu Koskensilta on 20 May 2010 17:48
George Greene <greeneg(a)email.unc.edu> writes:

> So we cannot prove that if there is a model, then there is a standard
> model?

Right.

> "Standard" here means well-founded and what else?

Nothing else, really.

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

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frode Bjørdal on 24 May 2010 13:53
On 20 Mai, 11:00, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(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.

Of course not.



> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 24 May 2010 13:59
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes:

> On 20 Mai, 11:00, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
>> The least alpha such that L_alpha is a model of set theory is not a
>> recursive ordinal.
>
> Of course not.

Then why did you mention "unprovable ordinals"?

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

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frode Bjørdal on 24 May 2010 14:15
On 20 Mai, 11:04, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(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.

Yes, the suggestion needs to be spelled out in more detail. This
cannot, I believe, be done satisfactorily here, as if in a tennis
match. :) Please take note that my suggestion was not limited to the
aleph-one of the minimal model. I was more precise.

There are, as many will agree on, several reasons to be dissatisfied
with the ZF-picture. The essential non-absoluteness phenomena
connected
with it is one. Another one, and the one I was invoking, is that we do
not have a safe enough semantics for the system. To invoke a
"cumulative
hierarchy" is, to my mind, ontologically extravagant. I believe in
Gödel's constructible hierarchy, but why should I believe there are so
large countable ordinals that we at some point get a model of ZF? I
seriously doubt that there are so large ordinals.


> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Frode Bjørdal on 24 May 2010 14:27
On 24 Mai, 19:59, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(a)gmail.com> writes:
> > On 20 Mai, 11:00, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> >> The least alpha such that L_alpha is a model of set theory is not a
> >> recursive ordinal.
>
> > Of course not.
>
> Then why did you mention "unprovable ordinals"?
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

To quote myself:

If we follow
> > the scarequoted terminology of the Wikipedia entry on large countable
> > ordinals, this ordinal would be an "unprovable" ordinal..
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