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Single Idea 18115

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets ]

Full Idea

We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to.

Gist of Idea

We could add axioms to make sets either as small or as large as possible

Source

David Bostock (Philosophy of Mathematics [2009], 6.4)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.188

A Reaction

Bostock says most mathematicians reject the first option, and are undecided about the second option.

The 33 ideas with the same theme [general points about the basics of set theory]:

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]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
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]
17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
17796 There is a semi-categorical axiomatisation of set-theory [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]