David Bernier wrote: > The listings for terms beginning with the letter C can be found here: > http://members.aol.com/jeff570/c.html > > A quote from the entry for CATEGORICAL (axiom system) follows: > > ``Thus in his The Loss of Certainty (1980, p. 271) Morris Kline wrote: > > Older texts did "prove" that the basic systems were categorical; > (...) But the "proofs" were loose (...) No set > of axioms is categorical, despite "proofs" by Hilbert and others. > > This remark was corrected by C. Smorynski in an acrimonious review: > > The fact is, there are two distinct notions of axiomatics and, > with respect to one, the older texts did prove categoricity and not > merely "prove". > > [This entry was contributed by Carlos César de Araújo.] " > > One suspicion I have is that Smorynski's comment is related > to second order logic as distinguished from first order logic.
Yes. The categoricity proofs by Hilbert, Dedekind and others would today be said to establish the categoricity of certain second order theories. There are also categorical first order theories.
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"Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus