In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.[1] A theory in a formal system satisfying the principle of explosion is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences (informally "axioms") and the set of closed sentences provable from under some (specified, possibly implicitly) formal deductive system. The set of axioms is consistent when there is no formula such that and . A theory in a formal system in general is consistent if it proves everything; a consistent theory may have a provable pair of a sentence and its negation. A consistent theory is a syntactic notion, whose semantic counterpart is a satisfiable theory. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true.[2] This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.
In a sound formal system, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete.[citation needed] The completeness of the sentential calculus was proved by Paul Bernays in 1918[citation needed][3] and Emil Post in 1921,[4] while the completeness of predicate calculus was proved by Kurt Gödel in 1930,[5] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).[6] Stronger logics, such as second-order logic, are not complete.
A consistency proof is a mathematical proof that a particular theory is consistent.[7] The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by the incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).
Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.
Let be a signature, a theory in and a sentence in . We say that is a consequence of , or that entails , in symbols , if every model of is a model of . (In particular if has no models then entails .)(Please note the definition of Mod(T) on p. 30 ...)
Warning: we don't require that if then there is a proof of from . In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use to mean that is deducible from in some particular formal proof calculus, and they write for our notion of entailment (a notation which clashes with our ). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.
We say that is valid, or is a logical theorem, in symbols , if is true in every -structure. We say that is consistent if is true in some -structure. Likewise, we say that a theory is consistent if it has a model.
We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T).
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