72 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] |
| 9456 | Modal logic is multiple systems, shown in the variety of accessibility relations between worlds [Jacquette] |
| 7689 | The modal logic of C.I.Lewis was only interpreted by Kripke and Hintikka in the 1960s [Jacquette] |
| 8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| 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] |
| 9457 | The two main views in philosophy of logic are extensionalism and intensionalism [Jacquette] |
| 7681 | Logic describes inferences between sentences expressing possible properties of objects [Jacquette] |
| 9463 | Classical logic is bivalent, has excluded middle, and only quantifies over existent objects [Jacquette] |
| 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] |
| 7682 | Logic is not just about signs, because it relates to states of affairs, objects, properties and truth-values [Jacquette] |
| 7697 | On Russell's analysis, the sentence "The winged horse has wings" comes out as false [Jacquette] |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| 9466 | Nominalists like substitutional quantification to avoid the metaphysics of objects [Jacquette] |
| 9465 | Substitutional universal quantification retains truth for substitution of terms of the same type [Jacquette] |
| 10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
| 9458 | Extensionalists say that quantifiers presuppose the existence of their objects [Jacquette] |
| 9461 | Intensionalists say meaning is determined by the possession of properties [Jacquette] |
| 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] |
| 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] |
| 7701 | Can a Barber shave all and only those persons who do not shave themselves? [Jacquette] |
| 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] |
| 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] |
| 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] |
| 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] |
| 7707 | To grasp being, we must say why something exists, and why there is one world [Jacquette] |
| 7687 | Existence is completeness and consistency [Jacquette] |
| 7692 | Being is maximal consistency [Jacquette] |
| 7679 | Ontology is the same as the conceptual foundations of logic [Jacquette] |
| 7678 | Ontology must include the minimum requirements for our semantics [Jacquette] |
| 7683 | Logic is based either on separate objects and properties, or objects as combinations of properties [Jacquette] |
| 7684 | Reduce states-of-affairs to object-property combinations, and possible worlds to states-of-affairs [Jacquette] |
| 7703 | If classes can't be eliminated, and they are property combinations, then properties (universals) can't be either [Jacquette] |
| 7685 | An object is a predication subject, distinguished by a distinctive combination of properties [Jacquette] |
| 7699 | Numbers, sets and propositions are abstract particulars; properties, qualities and relations are universals [Jacquette] |
| 7691 | The actual world is a consistent combination of states, made of consistent property combinations [Jacquette] |
| 7688 | The actual world is a maximally consistent combination of actual states of affairs [Jacquette] |
| 7695 | Do proposition-structures not associated with the actual world deserve to be called worlds? [Jacquette] |
| 7694 | We must experience the 'actual' world, which is defined by maximally consistent propositions [Jacquette] |
| 2975 | That honey is sweet I do not affirm, but I agree that it appears so [Timon] |
| 7706 | If qualia supervene on intentional states, then intentional states are explanatorily fundamental [Jacquette] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 7704 | Reduction of intentionality involving nonexistent objects is impossible, as reduction must be to what is actual [Jacquette] |
| 9460 | Extensionalist semantics forbids reference to nonexistent objects [Jacquette] |
| 9459 | Extensionalist semantics is circular, as we must know the extension before assessing 'Fa' [Jacquette] |
| 7702 | The extreme views on propositions are Frege's Platonism and Quine's extreme nominalism [Jacquette] |