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] |
| 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] |
| 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] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 21181 | Relativity and Quantum theory give very different accounts of forces [Hesketh] |
| 21183 | Thermodynamics introduced work and entropy, to understand steam engine efficiency [Hesketh] |
| 21191 | Photons are B and W° bosons, linked by the Higgs mechanism [Hesketh] |
| 21199 | Spinning electric charge produces magnetism, so all fermions are magnets [Hesketh] |
| 21189 | Electrons may have smaller components, bound by a new force [Hesketh] |
| 21180 | Electrons are fundamental and are not made of anything; they are properties without size [Hesketh] |
| 21182 | Quantum mechanics is our only theory, and is very precise, and repeatedly confirmed [Hesketh] |
| 21184 | Physics was rewritten to explain stable electron orbits [Hesketh] |
| 21187 | Virtual particles can't be measured, and can ignore the laws of physics [Hesketh] |
| 21185 | Colour charge is positive or negative, and also has red, green or blue direction [Hesketh] |
| 21194 | The Standard Model omits gravity, because there are no particles involved [Hesketh] |
| 21195 | In Supersymmetry the Standard Model simplifies at high energies [Hesketh] |
| 21197 | Standard Model forces are one- two- and three-dimensional [Hesketh] |
| 21188 | Quarks and leptons have a weak charge, for the weak force [Hesketh] |
| 21186 | Quarks rush wildly around in protons, restrained by the gluons [Hesketh] |
| 21192 | Neutrinos only interact with the weak force, but decays produce them in huge numbers [Hesketh] |
| 21196 | To combine the forces, they must all be the same strength at some point [Hesketh] |
| 21190 | 'Space' in physics just means location [Hesketh] |
| 21193 | The universe is 68% dark energy, 27% dark matter, 5% regular matter [Hesketh] |
| 21198 | If a cosmic theory relies a great deal on fine-tuning basic values, it is probably wrong [Hesketh] |