⊿ Point. {R} Glossary. ◢ Keyword: C. ◥ University. {q} PhD. {tr} Training. ⚫ UK. ↂ EndNote. ✊ Harvey (2009) Consistency. Non-contradiction of a theory 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. Is consistent when there is no formula such that and A consistency proof is a mathematical proof that a particular theory is consistent.[8] 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.
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. Consistency and completeness in arithmetic and set theory [edit] Presburger arithmetic is an axiom system for the natural numbers under addition. In the following context of mathematical logic, the turnstile symbol means "provable from".
Let Define an equivalence relation.