28 ideas
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
7548 | Classes, grouped by a convenient property, are logical constructions [Russell] |
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] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
7545 | Visible things are physical and external, but only exist when viewed [Russell] |
7553 | Sense-data are purely physical [Russell] |
7549 | If my body literally lost its mind, the object seen when I see a flash would still exist [Russell] |
7546 | A man is a succession of momentary men, bound by continuity and causation [Russell] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
7550 | We could probably, in principle, infer minds from brains, and brains from minds [Russell] |
7547 | Matter requires a division into time-corpuscles as well as space-corpuscles [Russell] |
7551 | Matter is a logical construction [Russell] |
7552 | Six dimensions are needed for a particular, three within its own space, and three to locate that space [Russell] |
20713 | God must be fit for worship, but worship abandons morally autonomy, but there is no God [Rachels, by Davies,B] |