[FOM] Hilbert and conservativeness
Robert Black
Mongre at gmx.de
Sat Sep 3 07:32:07 EDT 2005
Panu Raatikainen wrote:
>My hypothesis is the following: I think that Hilbert simply assumed
>that finitistic mathematics is deductively complete with respect to the
>real sentences (i.e., is “real-complete”).... Bernays, in any case,
>explicitly assumed
>real-completeness: “In the case of a finitistic proposition however, the
>determination of its irrefutability is equivalent to determination of its
>truth” (Bernays 1930, 259, my italics). One may presume that this also
>reliably reflects Hilbert’s view.
No doubt Hilbert (and Bernays) believed that finitary reasoning was
complete for real sentences. And this would follow anyway from other
things they believed, namely that PA was complete, that a finitary
proof of the consistency of PA was possible, and that such a proof
would show that PA was conservative over finitary reasoning for real
sentences.
However, to have assumed this in argument would have been a serious
mistake (and not one I think we should attribute to them), since the
Enemy was Brouwer, and Brouwer would (rightly, as it turns out) not
have conceded it.
The quote from Bernays just doesn't entail real completeness. From
the immediate context it's not even clear that he's talking about
*general* statements at all rather than just calculations with
particular numbers. But assume (I think probably correctly) that he
is talking about general statements. The sentence before tells us
that once we have recognized the consistency of an ideal system of
postulates 'it immediately follows that a theorem deduced from them
can never contradict an intuitively recognizable fact [anschaulich
erkennbare Tatsache]'. Note that the word 'Tatsache' would be more
naturally used for a *particular* fact than a general one. This
looks to me *exactly* like the argument of Hilbert's Hamburg lecture:
if AxFx is a theorem of a consistent system extending finitary
reasoning then there can't be an n such that not-Fn is an anschaulich
erkennbare Tatsache, so for every n Fn is an anschaulich erkennbare
Tatsache, so AxFx is true. From the finitary standpoint AxFx is
incapable of negation, so the only sense in which it could be
refutable is for there to be an n such that not-Fn calculates out as
true, and for it to be irrefutable just is for it to be the case that
for every n, Fn calculates out as true. Nothing about real
completeness here.
Note also that intuitionistically (in his 1930 paper Bernays, as he
later noted, didn't distinguish properly between 'finitary' and
'intutionistic') if F is decidable, then from not-not-AxFx we can
conclude AxFx, i.e. stability but not decidability holds for pi_1
sentences.
Robert
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