17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
17879 | Axiomatising set theory makes it all relative [Skolem] |
13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
9565 | Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara] |
3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn] |
8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
9879 | NF has no models, but just blocks the comprehension axiom, to avoid contradictions [Quine, by Dummett] |
9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
10191 | Set theory reduces to a mereological theory with singletons as the only atoms [Lewis, by MacBride] |
15507 | Set theory has some unofficial axioms, generalisations about how to understand it [Lewis] |
10073 | There cannot be a set theory which is complete [Smith,P] |
3335 | The standard Z-F Intuition version of set theory has about ten agreed axioms [Benardete,JA, by PG] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Hallett,M] |
17833 | The first-order ZF axiomatisation is highly non-categorical [Hallett,M] |
17834 | Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Hallett,M] |
13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Clegg] |
10653 | Maybe set theory need not be well-founded [Varzi] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
18843 | The iterated conception of set requires continual increase in axiom strength [Rumfitt] |