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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]
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13494
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The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
[Hart,WD]
|
17801
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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]
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