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Single Idea 17608

[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory ]

Full Idea

Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory.

Gist of Idea

We take set theory as given, and retain everything valuable, while avoiding contradictions

Source

Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)

Book Ref

'From Frege to Gödel 1879-1931', ed/tr. Heijenoort,Jean van [Harvard 1967], p.200

A Reaction

Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets.

The 14 ideas from 'Investigations in the Foundations of Set Theory I'

13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
9627 Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
15924 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
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]
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]