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In mathematical logic, an ω-consistent (or omega-consistent) theory is a theory (collection of sentences) that is not only consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. Specifically, if T is a theory that interprets arithmetic (that is, there is a way to understand some of its objects of discourse as natural numbers), then T is ω-inconsistent if, for some property P of natural numbers (definable in the language of T), T proves P(0), P(1), P(2), and so on (that is, for every natural number n, T proves that P(n) holds), but T also proves that there is some natural number n such that P(n) fails. This may not lead directly to an outright contradiction, because T may not be able to prove for any specific value of n that P(n) fails, only that there is such an n. T is ω-consistent if it is not ω-inconsistent.
Example Write PA for the theory Peano arithmetic, and Con(PA) for the statement of arithmetic that formalizes the claim "PA is consistent". Con(PA) could be of the form "For every natural number n, n is not the Gödel number of a proof from PA that 0=1". (This formulation uses 0=1 instead of a direct contradiction; that gives the same result, because PA certainly proves ¬0=1, so if it proved 0=1 as well we would have a contradiction, and on the other hand, if PA proves a contradiction, then it proves anything, including 0=1.) Now, assuming PA is really consistent, it follows that PA+¬Con(PA) is also consistent, for if it were not, then PA would prove Con(PA), contradicting Gödel's second incompleteness theorem. However, PA+¬Con(PA) is not ω-consistent. This is because, for any particular natural number n, PA+¬Con(PA) proves that n is not the Gödel number of a proof that 0=1 (PA itself proves that fact; the extra assumption ¬Con(PA) is not needed). However, PA+¬Con(PA) proves that, for some natural number n, n is the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA) ). Relation to other consistency principles If the theory T is recursively axiomatizable, ω-consistency has the following characterization, due to C. Smoryński: T is ω-consistent if and only if is consistent. Here, is the set of all Π02-sentences valid in the standard model of arithmetic, and is the uniform reflection principle for T, which consists of the axioms for every formula with one free variable. In particular, a finitely axiomatizable theory T in the language of arithmetic is ω-consistent if and only if T + PA is -sound. | ||||||||
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