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Full Idea
Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency.
Gist of Idea
Second Incompleteness: nice theories can't prove their own consistency
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.6
A Reaction
On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'.
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 13341 | Using the definition of truth, we can prove theories consistent within sound logics [Tarski] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 12656 | P-and-Q gets its truth from the truth of P and truth of Q, but consistency isn't like that [Fodor] |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| 18785 | Consistency is semantic, but non-contradiction is syntactic [Mares] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |