33 ideas
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
18806 | Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
8492 | Relations are functions with two arguments [Frege] |
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] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
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] |
8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
18899 | Frege takes the existence of horses to be part of their concept [Frege, by Sommers] |
4028 | Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege] |
8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
9947 | Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman] |
10319 | An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale] |
8488 | A concept is a function whose value is always a truth-value [Frege] |
9948 | Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman] |
4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege] |
21947 | Power is localised, so we either have totalitarian centralisation, or local politics [Foucault, by Gutting] |
21946 | Prisons gradually became our models for schools, hospitals and factories [Foucault, by Gutting] |
8491 | The Ontological Argument fallaciously treats existence as a first-level concept [Frege] |