15653 | We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner] |
10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] |
10067 | Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave] |
10554 | Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |