[FOM] Hilbert and conservativeness

[Robert Black]

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