consistency in traditional Aristotelian logic, a semantic notion: two or more statements are called consistent if they are simultaneously true under some interpretation (cf., e.g., W. S. Jevons, Elementary Lessons in Logic, 1870). In modern logic there is a syntactic definition that also fits complex (e.g., mathematical) theories developed since Frege’s Begriffsschrift (1879): a set of statements is called consistent with respect to a certain logical calculus, if no formula ‘P & –P’ is derivable from those statements by the rules of the calculus; i.e., the theory is free from contradictions. If these definitions are equivalent for a logic, we have a significant fact, as the equivalence amounts to the completeness of its system of rules. The first such completeness theorem was obtained for sentential or propositional logic by Paul Bernays in 1918 (in his Habilitationsschrift that was partially published as Axiomatische Untersuchung des Aussagen-Kalküls der ‘Principia Mathematica,’ 1926) and, independently, by Emil Post (in Introduction to a General Theory of Elementary Propositions, 1921); the completeness of predicate logic was proved by Gödel (in Die Vollständigkeit der Axiome des logischen Funktionenkalküls, 1930). The crucial step in such proofs shows that syntactic consistency implies semantic consistency. Cantor applied the notion of consistency to sets. In a well-known letter to Dedekind (1899) he distinguished between an inconsistent and a consistent multiplicity; the former is such ‘that the assumption that all of its elements ‘are together’ leads to a contradiction,’ whereas the elements of the latter ‘can be thought of without contradiction as ‘being together.’ ‘ Cantor had conveyed these distinctions and their motivation by letter to Hilbert in 1897 (see W. Purkert and H. J. Ilgauds, Georg Cantor, 1987). Hilbert pointed out explicitly in 1904 that Cantor had not given a rigorous criterion for distinguishing between consistent and inconsistent multiplicities. Already in his Über den Zahlbegriff (1899) Hilbert had suggested a remedy by giving consistency proofs for suitable axiomatic systems; e.g., to give the proof of the ‘existence of the totality of real numbers or – in the terminology of G. Cantor – the proof of the fact that the system of real numbers is a consistent (complete) set’ by establishing the consistency of an axiomatic characterization of the reals – in modern terminology, of the theory of complete, ordered fields. And he claimed, somewhat indeterminately, that this could be done ‘by a suitable modification of familiar methods.’ After 1904, Hilbert pursued a new way of giving consistency proofs. This novel way of proceeding, still aiming for the same goal, was to make use of the formalization of the theory at hand. However, in the formulation of Hilbert’s Program during the 1920s the point of consistency proofs was no longer to guarantee the existence of suitable sets, but rather to establish the instrumental usefulness of strong mathematical theories T, like axiomatic set theory, relative to finitist mathematics. That focus rested on the observation that the statement formulating the syntactic consistency of T is equivalent to the reflection principle Pr(a, ‘s’) P s; here Pr is the finitist proof predicate for T, s is a finitistically meaningful statement, and ‘s’ its translation into the language of T. If one could establish finitistically the consistency of T, one could be sure – on finitist grounds – that T is a reliable instrument for the proof of finitist statements.
There are many examples of significant relative consistency proofs: (i) non-Euclidean geometry relative to Euclidean, Euclidean geometry relative to analysis; (ii) set theory with the axiom of choice relative to set theory (without the axiom of choice), set theory with the negation of the axiom of choice relative to set theory; (iii) classical arithmetic relative to intuitionistic arithmetic, subsystems of classical analysis relative to intuitionistic theories of constructive ordinals. The mathematical significance of relative consistency proofs is often brought out by sharpening them to establish conservative extension results; the latter may then ensure, e.g., that the theories have the same class of provably total functions. The initial motivation for such arguments is, however, frequently philosophical: one wants to guarantee the coherence of the original theory on an epistemologically distinguished basis.
See also CANTOR, COMPLETENESS, GÖDEL’S INCOMPLETENESS THEOREMS , HILBERT ‘ S PRO – GRAM , PROOF THEORY. W.S.