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Re: Matheology § 038
Posted:
Jun 18, 2012 4:36 PM
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On Jun 18, 5:11 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Gus Gassmann" <horand.gassm...@googlemail.com> wrote in message
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> news:8f33566a-f2a1-4aef-811d-b7e8a447168c@j9g2000vbk.googlegroups.com...
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> > On Jun 18, 1:22 pm, Virgil <vir...@ligriv.com> wrote:
> >> In article
> >> <3694188a-d258-4273-b904-c9dc8ce80...@f14g2000yqe.googlegroups.com>,
> >> WM <mueck...@rz.fh-augsburg.de> wrote:
> >> > On 17 Jun., 19:36, PotatoSauce <kiwisqu...@gmail.com> wrote:
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> >> > > Why are we talking about set of all sets?
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> >> > Because your Platonism implies that problem unless you errect an
> >> > artificial border.
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> >> It is abuse of language called self-reference that produces those
> >> problems, not the set theory.
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> >> ZFC, NBG, Morse-Kelley and similar set theories, have no such problems.
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> > Wait a minute. Am I parsing this correctly? ZFC has no problem with
> > self-reference? I thought Goedel showed that any axiom system strong
> > enough to encode PA can be made to be self-referential. Isn't that the
> > essence of the Goedel sentence and in fact of the proof?
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> Maybe it is not too incorrect to say (informally) that what is required is
> an arithmetical system at least strong enough to be able to describe its own
> behavior (i.e. where one can code statements and have symbols to talk about
> the set of -say- provable/refutable sentences, etc. But I won't guarantee
> for the details...).
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> Otherwise no, self-referentiality is not a necessary ingredient to construct
> undecidable statements:
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> <http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#The...>
Interesting. I'll have to try to digest that. Thanks.
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