Consistency proof

In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition φ are both φ and ¬φ provable. A consistency proof is a formal proof that a formal system is consistent. 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 fell to Gödel's insight, as expressed in his two incompleteness theorems, that sufficiently strong proof theories cannot prove their own consistency.
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