Consistency

In mathematics, a formal system is said to be consistent if none of its proven theorems can also be disproven within that system. Or, alternatively, if the formal system does not assign both true and false as the semantics of one given statement. These are definitions in negative terms - they speak about the absence of inconsistency. Formal systems that do admit contradictions suffer a semantic collapse, in the sense that deductions in them cannot truly be assigned any significant content, by schemes that apply across the whole system.

To add:

• Systems proved to be consistent
  • Presburger arithmetic
• Systems not proved consistent
  • First order Peano arithmetic (from a system no stronger than Peano arithmetic)
• Systems that cannot be proved consistent

See also:

• Gödel's incompleteness theorem
• consistency (statistics)