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| Aatu Koskensilta... |
 Posted: Fri Oct 30, 2009 2:59 pm |
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Rupert <rupertmccallum at (no spam) yahoo.com> writes:
[quote]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.
[/quote]
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 at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Sergei Tropanets... |
| Posted: Tue Nov 17, 2009 6:25 am |
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On Oct 30, 11:55 pm, Marina Gotovchits <renessa... at (no spam) gmail.com> wrote:
[quote]On 30 Okt, 21:59, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
Rupert <rupertmccal... at (no spam) 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?)
[/quote]
I think this your bewilderment is the result of applying such a
meaningless notions as "the set of all true arithmetical
sentences" (or "arithmetical truths"). What is this? For sure, even
set theorists working in classical logic can not define what those
words mean. As a result - theories like WO with strange set of axioms.
I think it is more reasonable to talk about decidable arithmetical
sentences in one or another sense.
Sergei Tropanets |
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