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Single Idea 13019

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

Full Idea

The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B.

Gist of Idea

The Iterative Conception says everything appears at a stage, derived from the preceding appearances

Source

Penelope Maddy (Believing the Axioms I [1988], §1.3)

Book Ref

-: 'Journal of Symbolic Logic' [-], p.485

A Reaction

Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story?

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]