FOM: June 1 - June 24, 1999

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FOM: more on Hilbert and Goedel



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  



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