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