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*To*: dubucs@idf.ext.jussieu.fr, fom@math.psu.edu*Subject*: FOM: more on Hilbert and Goedel*From*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>*Date*: Mon, 14 Jun 1999 08:37:28 -0400 (EDT)*Cc*: neilt@mercutio.cohums.ohio-state.edu*Sender*: owner-fom@math.psu.edu

In answer to my question _______________ Is there any clue in Hilbert's early writings that he countenanced the following possibility, with respect to a sentence S and a given theory?: (1) the effective test (for the presence of property P) shows that S is not provable; and (2) the effective test shows that not-S is not provable; while yet (3) the logic underlying the theory is known to be complete (i.e. every logical consequence admits of finitary proof) ? _______________ Jacquest Dubucs writes (Mon Jun 14 04:32:50 1999) _______________ I suspect that there is no such possible clue, for a simple and well-known reason. As lenthly and convincingly explained by Goedel in several places, the very notions in terms of which your (3) is expressed were not available to Hilbert, who was always very reluctant (and, by the way, unable) to formulate the completness problems in such semantical terms. Moreover, one can wonder if Hilbert's views favouring decidability were not a simple by-product of his propensity of thinking the foundational problems in exclusively syntactical terms (cf my posting of Jun 11 22:34:58 1999). _______________ 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. Did Goedel, in any of the lengthy and convincing explanations you refer to above, *show* (as opposed to: merely *claim*) that Hilbert was innocent of the completeness problem for first-order logic? It would be hard to see how he could. First, Goedel would have needed to know that there was no textual evidence, of the kind Sieg has come across in the Hilbert Nachlass, for the claim that Hilbert did not conceive of the completeness of first-order logic in adequate terms. Secondly, given the great embarrassment, for Hilbert, of Goedel's incompleteness result, he (H.) would be highly unlikely to have emphasized the actual clarity with which he conceived of logical completeness back in 1917, yet failed to appreciate that it could be a different thing from theory-completeness. One need only read the account of the way Hilbert's nose was put out of joint by Goedel's result (see Dawson's biography of Goedel) to appreciate this psychological claim. Neil Tennant

**Follow-Ups**:**dubucs**- FOM: Re: more on Hilbert and Goedel

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