Editing Gödel's incompleteness theorems (section)

=== Consistency ===
A set of axioms is (simply) [[Consistency|consistent]] if there is no statement such that both the statement and its negation are provable from the axioms, and ''inconsistent'' otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction.

Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an [[inaccessible cardinal]]" proves ZFC is consistent because if {{mvar|κ}} is the least such cardinal, then {{math|''V''<sub>{{mvar|κ}}</sub>}} sitting inside the [[von Neumann universe]] is a [[Inner model|model]] of ZFC, and a theory is consistent if and only if it has a model.

If one takes all statements in the language of [[Peano arithmetic]] as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.

Additional examples of inconsistent theories arise from the [[Naïve set theory#Paradoxes in early set theory|paradoxes]] that result when the [[Axiom schema of specification#Unrestricted comprehension|axiom schema of unrestricted comprehension]] is assumed in set theory.