59 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] |
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] |
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] |
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] |
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] |
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] |
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] |
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] |
17536 | If it can't be expressed mathematically, it can't occur in nature? [Heisenberg] |
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] |
17545 | Quantum theory shows that exact science does not need dogmatic realism [Heisenberg] |
17538 | Quantum theory does not introduce minds into atomic events [Heisenberg] |
17534 | A 'probability wave' is a quantitative version of Aristotle's potential, a mid-way type of reality [Heisenberg] |
17553 | We can retain the idea of 'substance', as indestructible mass or energy [Heisenberg] |
17544 | Basic particles have a mathematical form, which is more important than their substance [Heisenberg] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
17550 | We give a mathematical account of a system of natural connections in order to clarify them [Heisenberg] |
6552 | You can only explain the qualities of large objects using entities which lack those qualities [Heisenberg] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
17549 | Seven theories in science: mechanics, heat, electricity, quantum, particles, relativity, life [Heisenberg, by PG] |
17540 | Energy is that which moves, and is the substance from which everything is made [Heisenberg] |
17541 | Energy is an unchanging substance, having many forms, and causing all change [Heisenberg] |
17548 | Maxwell introduced real fields, which transferred forces from point to point [Heisenberg] |
17533 | Radiation interference needs waves, but radiation photoelectric effects needs particles [Heisenberg] |
17532 | An atom's stability after collisions needs explaining (which Newton's mechanics can't do) [Heisenberg] |
17537 | Position is complementary to velocity or momentum, so the whole system is indeterminate [Heisenberg] |
17551 | It was formerly assumed that electromagnetic waves could not be a reality in themselves [Heisenberg] |
17543 | So-called 'empty' space is the carrier of geometry and kinematics [Heisenberg] |
17552 | In relativity the length of the 'present moment' is relative to distance from the observer [Heisenberg] |