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Single Idea 10484

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

Full Idea

According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage.

Gist of Idea

The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first

Source

George Boolos (Must We Believe in Set Theory? [1997], p.126)

Book Ref

Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.126

A Reaction

He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent.

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]