27 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] |
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] |
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] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
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] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
472 | No things would be clear to us as entity or relationships unless there existed Number and its essence [Philolaus] |
19696 | There are reasons 'for which' a belief is held, reasons 'why' it is believed, and reasons 'to' believe it [Neta] |
19697 | The basing relation of a reason to a belief should both support and explain the belief [Neta] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
1518 | Everything must involve numbers, or it couldn't be thought about or known [Philolaus] |
1519 | Harmony must pre-exist the cosmos, to bring the dissimilar sources together [Philolaus] |
473 | There is no falsehood in harmony and number, only in irrational things [Philolaus] |
469 | Existing things, and hence the Cosmos, are a mixture of the Limited and the Unlimited [Philolaus] |
476 | Self-created numbers make the universe stable [Philolaus] |
1787 | Philolaus was the first person to say the earth moves in a circle [Philolaus, by Diog. Laertius] |