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Single Idea 10565

[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets ]

Full Idea

There is no stage at which we can take all the sets to have been generated, since the set of all those sets which have been generated at a given stage will itself give us something new.

Gist of Idea

There is no stage at which we can take all the sets to have been generated

Source

Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)

Book Ref

-: 'Philosophical Studies' [-], p.372

The 15 ideas with the same theme [sets as a well-founded hierarchy built from scratch]:

10484 The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos]
13494 The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD]
17801 The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
10565 There is no stage at which we can take all the sets to have been generated [Fine,K]
13019 The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy]
13640 Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
9617 The 'iterative' view says sets start with the empty set and build up [Brown,JR]
10405 In the iterative conception of sets, they form a natural hierarchy [Swoyer]
10708 Nowadays we derive our conception of collections from the dependence between them [Potter]
15900 The iterative conception of set wasn't suggested until 1947 [Lavine]
15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
15932 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
23624 The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack]