64 ideas
| 17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| 17751 | Gödel proved the completeness of first order predicate logic in 1930 [Gödel, by Walicki] |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| 8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
| 9942 | Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam] |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| 18151 | Could we replace sets by the open sentences that define them? [Chihara, by Bostock] |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| 8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
| 9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett] |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| 10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
| 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] |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
| 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] |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| 18062 | Set-theory paradoxes are no worse than sense deception in physics [Gödel] |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
| 10868 | The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg] |
| 13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD] |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| 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] |
| 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] |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| 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] |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| 10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
| 10265 | Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro] |
| 8759 | We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro] |
| 10264 | Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro] |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| 594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| 2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |