15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Lavine] |
17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions |
17607 | Set theory investigates number, order and function, showing logical foundations for mathematics |
13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Maddy] |
17609 | Set theory can be reduced to a few definitions and seven independent axioms |
10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Clegg] |
9565 | Zermelo made 'set' and 'member' undefined axioms [Chihara] |
3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Blackburn] |
17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Hallett,M] |
13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Maddy] |
13028 | Replacement was added when some advanced theorems seemed to need it [Maddy] |
13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Maddy] |
13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Maddy] |
13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Hart,WD] |
17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers |
13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Hart,WD] |
15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Lavine] |
18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Maddy] |
13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Maddy] |
9627 | Different versions of set theory result in different underlying structures for numbers [Brown,JR] |
17613 | We should judge principles by the science, not science by some fixed principles |