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From: Marina Gotovchits on 30 Oct 2009 13:24
On 30 Okt, 17:56, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Oct 30, 9:41 pm, Marina Gotovchits <renessa...(a)gmail.com> wrote:
>
>
>
>
>
> > On 30 Okt, 00:28, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> > > Marina Gotovchits <renessa...(a)gmail.com> writes:
> > > > My question should have been posed as folllows, perhaps: If we have
> > > > the omega-rula along the climb up the hierarchy of subsystems os SOA,
> > > > how far do we need to go in order to have a model of ZF if ZF is
> > > > consistent?
>
> > > WKL-0 proves the completeness theorem, and thus in particular that if ZF
> > > is consistent it has a model.
>
> > Aha! Thank you very much Aatu!! I recall Simpson's Theorem I.10.3.8
> > (page 36 of his Book), which states that WKL(0) is equivalent, over RCA
> > (0), to "Gödel's completeness theorem: every (consistent) finite, or
> > countable, set of sentences in the predicate calculus has a countable
> > model.
>
> > Does this mean that if ZF is consistent, then a countable model "lives
> > in" WKL(0)+the omega rule. Or do we need to go slightly higher for the
> > model to be representable by a "set" in a subsystem of SOA. I ask this
> > because Gödel's completeness theorem here may be interpreted
> > existentially relative to WKL(0). I.e.. GCT and WKL(0) are equivalent,
> > over RCA(0), so maybe WKL(0)+ the omega rule only asserts that there
> > IS a countable model of ZF (if ZF is consistent), without itself
> > exhibiting such a model. If so, when does the model itself become a
> > "set"?
>
> Assuming that ZF is consistent, in WKL_0+the omega rule we can prove
> that there exists a model of ZF. We probably can't prove that the
> natural numbers are standard in this model. Was that your question?

No.I thought the omega rule would guarantee that the natural numbers
are standard. Am I wrong?

Let WO be the system WKL(0) + the omega-rule. From what's on the
table, as I understand it, WO prowes that (EX)(X is a model of ZF).
Here the quantifier is over sets, and not numbers, of course. The
question is wheter there, in case ZF is consistent, is a closed set-
term A such that WO proves that A is a model of ZF.




From: Marina Gotovchits on 30 Oct 2009 14:50

> Let WO be the system WKL(0) + the omega-rule. From what's on the
> table, as I understand it, WO prowes that (EX)(X is a model of ZF).
> Here the quantifier is over sets, and not numbers, of course. The
> question is wheter there, in case ZF is consistent, is a closed set-
> term A such that WO proves that A is a model of ZF.

I now realize that SOA of course does not have the kind of closed set
terms I was here asking for, so that (EX)(X is a model of ZF) is as
much as we can get in SOA. (My questions are motivated from another
context.) Anyway, intuitively this says that a model of ZF exists.

Is the model transitive?
From: Aatu Koskensilta on 30 Oct 2009 16:59
Rupert <rupertmccallum(a)yahoo.com> writes:

> Assuming that ZF is consistent, in WKL_0+the omega rule we can prove
> that there exists a model of ZF. We probably can't prove that the
> natural numbers are standard in this model.

We can't prove in WKL_0 + all arithmetical truths that WKL_0 + all
arithmetical truths is consistent, and hence certainly not that ZFC +
all arithmetical truths is consistent (which is just another way of
saying: ZFC has an omega-model).

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

"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Rupert on 30 Oct 2009 17:13
On Oct 31, 5:50 am, Marina Gotovchits <renessa...(a)gmail.com> wrote:
> > Let WO be the system WKL(0) + the omega-rule. From what's on the
> > table, as I understand it, WO prowes that (EX)(X is a model of ZF).
> > Here the quantifier is over sets, and not numbers, of course. The
> > question is wheter there, in case ZF is consistent, is a closed set-
> > term A such that WO proves that A is a model of ZF.
>
> I now realize that SOA of course does not have the kind of closed set
> terms I was here asking for, so that (EX)(X is a model of ZF) is as
> much as we can get in SOA. (My questions are motivated from another
> context.) Anyway, intuitively this says that a model of ZF exists.
>
> Is the model transitive?

We can prove that a model exists, but we cannot prove that a well-
founded model exists, for the reasons Aatu has discussed.
From: Marina Gotovchits on 30 Oct 2009 17:55
On 30 Okt, 21:59, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Rupert <rupertmccal...(a)yahoo.com> writes:
> > Assuming that ZF is consistent, in WKL_0+the omega rule we can prove
> > that there exists a model of ZF. We probably can't prove that the
> > natural numbers are standard in this model.
>
> We can't prove in WKL_0 + all arithmetical truths that WKL_0 + all
> arithmetical truths is consistent, and hence certainly not that ZFC +
> all arithmetical truths is consistent (which is just another way of
> saying: ZFC has an omega-model).

I am a bit bewildered at this point. We agreed that if ZF is
consistent, then an arithmetical sentence CON(ZF) will hold true, and
so is provable in WKL(0)+the omega-rule(=WO in the following). An
appeal to the fact that WKL(0) is equivalent, under RCA(0), to Gödel's
completeness theorem, then licenses the inference that WO proves that
there is a model of ZF.

OF course, WO cannot prove its own consistency. But is CON(WO) at all
a well-formed sentence? For a recursively axiomatizable theory like ZF
we indeed have an arithmetical provability predicate and so can define
CON(ZF). But it seems to me that we do not have an arithmetical
provability predicate for WO. So it is not clear to my mind what the
statement "WO is consistent" should mean in arithmetical terms, and
how it relates to my starting point with CON(ZF).

In my original query I also related a similar query to the 1-
consistency of certain strong theories. Let us strengthen this to
their omega consistency. We may as well concentrate on ZF. Is not OCON
(ZF) (a statement to the effect that ZF is omega consistent) also
representable as an arithmetical sentence? If so, will it not produce
an omega consistent model e.g. in WO? (If it is not so representable,
there is something crucial in one at points heated discussion between
Harvey Friedman and Solomon Feferman which I miss out on. Is it that 1-
consistency is so representable while omega-consistency is not,
perhaps?)


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