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*To*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>*Subject*: FOM: Re: more on Hilbert and Goedel*From*: dubucs <dubucs@ext.jussieu.fr>*Date*: Mon, 14 Jun 1999 17:26:50 +0200*Cc*: fom@math.psu.edu*In-Reply-To*: <199906141237.IAA02547@mercutio.cohums.ohio-state.edu>*Sender*: owner-fom@math.psu.edu

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 IHPST CNRS Paris I 13, rue du Four 75006 Paris Tel (33) 01 43 54 60 36 (33) 01 43 54 94 60 Fax (33) 01 44 07 16 49

**Follow-Ups**:**Wilfried Sieg**- FOM: More on Hilbert (and Goedel)

**References**:**Neil Tennant**- FOM: more on Hilbert and Goedel

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