FOM: Re: more on Hilbert and Goedel

[dubucs]

Neil Tennant wrote:
>
>According to Wilfried Sieg, there is evidence in the Autumn
>1917/Winter 1918 lecture notes by Ackermann for Hilbert that the
>completeness problem for first-order predicate calculus was properly
>conceived, and set out as one to be settled positively. My suspicion
>is that what took a much longer time to dawn on people working in
>foundations back then was that common axiomatizations of well-known
>theories (such as Peano-Dedekind arithmetic) might be incomplete,
>despite being based on a complete logic.

I'm very curious and impatient to cast a glance at the lectures notes by
Ackermann that have been found by Sieg, for they seem able to change the
current view (at least my own!) concerning the completeness problem during
the twenties.

Note that, in a sense, such textual evidence make the historical enigma
about completeness more acute, rather than less: *why* in that case the
completeness theorem has not been proved before 1929, viewing that all the
mathematical pieces of the puzzle were available much earlier (cf Goedel's
letter to Wang from December 7, 1967, according which the theorem was an
"almost trivial consequence of Skolem 1922" (quoted in Wang's "From
Mathematics to Philosophy", p. 8)). Goedel's claim (of course not founded
on the evidence of the lack of any textual evidence of the contrary !) had
at least the merit of proposing a prima facie acceptable explanation of
this enigma, in terms of Hilbert's finitistic "prejudices" against
semantical notions.

Moreover, taking Sieg's windfalls in Hilbert's Nachlass into account, one
has now to explain also the "regress" of Hilbert towards a purely
syntactical formulation of the completeness problem in the late 1920's (cf
my posting of Jun 11 22:34:58). I suspect that any plausible explanation
should invoke Hilbert's decision of treating foundational questions as
strictly parallel to elementary number-theoretical questions, whence the
exclusion of the semantical notion of completeness. One has to add that the
partial decision results obtained about at the same time (Presburger, for
the theory of N as additive monoid) could have  convinced Hilbert that the
implications of his syntactical definition of completeness were, after all,
not too implausible.

JD




Jacques Dubucs
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