[FOM] Question on the Scope of Mathematics
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Thu Jul 29 02:14:15 EDT 2004
On Jul 27, 2004, at 9:46 PM, Henrik Nordmark wrote:
>
>> Aatu Koskensilta <aatu.koskensilta at xortec.fi> wrote:
>> But the only reason for assuming that provability in ZFC has anything
>> to do with
>> truth of statements is that ZFC is sound. By your account, this either
>> is not a mathematical
>> truth, or we should specify a formal system, say MK, in which it can
>> be
>> proved.
>
>
> Well, it depends on what you mean by a "mathematical truth".
> I sympathize with Dmytro's position in the sense that if one is
> working within the framework of ZFC, then the claim that ZFC is sound
> and is a good formalism lies at the meta-mathematical and not really
> at the mathematical level. Of course, one can go beyond the framework
> at hand and justify the claim that ZFC is sound by using stronger
> systems, but then again the issue of whether these stronger systems
> are any good is meta-mathematics.
While the question of whether ZFC is a good formalism might be
non-mathematical, I can't see why ZFC's soundness would not be a
mathematical question. In particular, we can express, in the language
of set theory, that ZFC is sound for sentences of complexity less
than n. Obviously we can't hope to prove these things in ZFC, since
they imply consistency of ZFC, but they can be expressed. The only
reason I can see to consider provability in ZFC relevant to
mathematical truth is belief in all of these partial soundness
assertions. For a particular sentence P one of these partial soundness
assertions suffices. According to Dmytro's position none of these
partial soundness assertions should coun't as mathematical truths. I
don't see why this doesn't mean that all of the set theoretical
sentences the truth of which is established by provability in ZFC
together with some partial soundness statement aren't mathematical
truths either.
Perhaps I'm just stuck with the idea that the truth of a statement A
should be equivalent to A, and thus they should both be equally
mathematical or non-mathematical and equally problematic or
non-problematic.
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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