32 ideas
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
9175 | We may fix the reference of 'Cicero' by a description, but thereafter the name is rigid [Kripke] |
9171 | The function of names is simply to refer [Kripke] |
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] |
17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
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] |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
9174 | It is necessary that this table is not made of ice, but we don't know it a priori [Kripke] |
9172 | A 'rigid designator' designates the same object in all possible worlds [Kripke] |
9173 | We cannot say that Nixon might have been a different man from the one he actually was [Kripke] |
9176 | Modal statements about this table never refer to counterparts; that confuses epistemology and metaphysics [Kripke] |
9177 | Identity theorists must deny that pains can be imagined without brain states [Kripke] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
9178 | Pain, unlike heat, is picked out by an essential property [Kripke] |