31 ideas
| 6564 | To affirm 'p and not-p' is to have mislearned 'and' or 'not' [Quine] |
| 11223 | Definitions usually have a term, a 'definiendum' containing the term, and a defining 'definiens' [Gupta] |
| 11215 | Notable definitions have been of piety (Plato), God (Anselm), number (Frege), and truth (Tarski) [Gupta] |
| 11227 | The 'revision theory' says that definitions are rules for improving output [Gupta] |
| 11225 | A definition needs to apply to the same object across possible worlds [Gupta] |
| 11221 | A definition can be 'extensionally', 'intensionally' or 'sense' adequate [Gupta] |
| 11224 | Traditional definitions are general identities, which are sentential and reductive [Gupta] |
| 11226 | Traditional definitions need: same category, mention of the term, and conservativeness and eliminability [Gupta] |
| 11217 | Chemists aim at real definition of things; lexicographers aim at nominal definition of usage [Gupta] |
| 11216 | If definitions aim at different ideals, then defining essence is not a unitary activity [Gupta] |
| 11218 | Stipulative definition assigns meaning to a term, ignoring prior meanings [Gupta] |
| 11220 | Ostensive definitions look simple, but are complex and barely explicable [Gupta] |
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| 11222 | The ordered pair <x,y> is defined as the set {{x},{x,y}}, capturing function, not meaning [Gupta] |
| 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| 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] |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |