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From: Marc Alcobé García on 5 May 2010 16:08
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?

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

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

3. Which is the least countable ordinal that can play that role?
From: Rupert on 5 May 2010 23:28
On May 6, 6:08 am, 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?
>

The universe of discourse contains all the objects which are ordinals
in the sense of the model.

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

The universe of discourse contains all the objects which are countable
ordinals in the sense of the model. There exists a bijection from the
set of all objects which are natural numbers in the sense of the model
to the set of all objects which are countable ordinals in the sense of
the model. But this bijection is not in the model.

> If the explanation of 2 is that simply the bijection between aleph_1
> and omega does not exist:
>
> 3. Which is the least countable ordinal that can play that role?

One thing you should bear in mind is that there are some nonstandard
models, that is, some models where the membership relation is not well-
founded when looked at from "outside" the model. In that case the
objects which are countable ordinals in the sense of the model cannot
necessarily be re-interpreted as corresponding to countable ordinals
when looked at from outside the model. 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.
From: Marc Alcobé García on 6 May 2010 10:17
On 6 mayo, 05:28, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On May 6, 6:08 am, 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?
>
> The universe of discourse contains all the objects which are ordinals
> in the sense of the model.
>
> > 2. How can that model contain all countable ordinals? (there are
> > uncountably many of them)
>
> The universe of discourse contains all the objects which are countable
> ordinals in the sense of the model. There exists a bijection from the
> set of all objects which are natural numbers in the sense of the model
> to the set of all objects which are countable ordinals in the sense of
> the model. But this bijection is not in the model.
>
> > If the explanation of 2 is that simply the bijection between aleph_1
> > and omega does not exist:
>
> > 3. Which is the least countable ordinal that can play that role?
>
> One thing you should bear in mind is that there are some nonstandard
> models, that is, some models where the membership relation is not well-
> founded when looked at from "outside" the model. In that case the
> objects which are countable ordinals in the sense of the model cannot
> necessarily be re-interpreted as corresponding to countable ordinals
> when looked at from outside the model. 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.

Yes, I was unconsciously assuming the countable models to be standard,
i. e. with the membership relation interpreted as real set membership.

Thank you for your answer.

Browsing through the Internet I have found a similar question:

http://mathoverflow.net/questions/16368/least-ordinal-not-in-a-countable-transitive-model-of-zfc
From: Aatu Koskensilta on 11 May 2010 09:37
Rupert <rupertmccallum(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.

--
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 12 May 2010 17:04
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..
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