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### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

#### [general points about the basics of set theory]

33 ideas
 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]
 17832 Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M]
 10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
 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]
 17835 Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
 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]
 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]
 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]
 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]