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*To*: Blind.Copy.Receiver@compuserve.com*Subject*: FOM: "Consistent":Three Questions.*From*: Robert Tragesser <RTragesser@compuserve.com>*Date*: Tue, 1 Dec 1998 09:55:59 -0500*Sender*: owner-fom@math.psu.edu

[Thanks to Davis, Tait, Poston, et al.; here I think is a slightly better framing of my concerns.] Three questions, one technical, two historical-philosophical, about "consistency". [1] TECHNICAL QUESTION. Is there an example of, or can one prove the existence of, a theory framed in a higher order classical logic [or at least a logic so understood that the class of semantical logical consequences and the class of syntactic consequences do not coincide; but hopefully the logic and the theory would be nontrivial wrt my question] that is (a) inconsistent, (b) a contradiction is not derivable. [2] HISTORICAL QUESTION 1. In German one has, _konsistent_ and _widerspruchslos_. I'm curious about the history of these terms, in the context of logic and/or mathematical demonstration. In GA III, Hilbert uses the second. (Well, I find _Widerspruchsfreiheit_ and _Widerspruchslosigkeit_; is there a difference in connotation?) I don't find 'konsistent'. Cantor speaks of "(in)konsistente Vielheit". In, e.g., _Cantor an Dedekind 28 Juli 1899_, Cantor has Omega' as "keine konsistente Vielheit" because it supports "ein Widerspruch". Is Cantor's use of 'konsistent' rather than 'widerspruchslos/frei' stylistic only, a matter of diction? In English one says "consistent" rather than "contradiction-free" because of the awkwardness of the latter; but one wonders if this doesn't sometimes conceal some important distinctions. [3] HISTORICAL QUESTION 2. This goes to the heart of my concern: a _reductio ad absurdu_m argument in the primary sense shows a proposition to be false by showing that it implies something "absurd". The latter meaning not only "false" but also "not true to" or even "dissonant with" (etc.). It does seem that "negation" in intuitionistic reasoning is most resonantly understood in these terms (reducible to something intuitionistically absurd; "false" for "absurd" won't do, of course). Here one then has a use for "consistent"--true to intuitionistic reasoning -- which only by slight of hand is explicable in terms of "contradiction-free". With classical logic, one is perhaps even required to restrict "absurdities" to contradictions; then of course one can use RAA arguments as proofs rather than as disproofs only (it is only by a kind of ruse that in intuitionistic reasoning what is essentially a disproof of p is made to appear to be a proof of "-p".). I am still wondering if for example supplying the upper half-plane model for Bolyai-Lobach. geometry does not _fundamentally_ (from the point of view of the mathematical achievement or effect) show B-L geometry to be "consistent" more in the sense of "consistent with mathematics", that is, it does bring B-L geometry fully into the fold of mathematics. Hanging out here (if I may speak colloquially) is the thought that in a fundamental sense, consistency proofs ought to lend mathematical signficance to a subject whose mathematical significance is in doubt. (Whereas one can worry over the significance of just supplying indirectly -- as in Go"del's completeness proof -- a model in the natural numbers that has no striking mathematical significance.) Of course, what I driving me is the great discomfort I've always felt with mathematics that is too logic driven, perhaps because I place a great value on understanding in some full or perhaps melodramatic sense. (The line about depriving a mathematician of the law of the excluded middle is like depriving a boxer of their fists has struck me as issuing from the sort of person who goes to boxing matches in the hopes of seeing a smashing knock-out punch, or listens to symphonies impatient for "the good parts".) Robert Tragesser

**Follow-Ups**:**Vladimir Sazonov**- Re: FOM: "Consistent":Three Questions.

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