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Single Idea 13640
[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
]
Full Idea
The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets.
Gist of Idea
Russell's paradox shows that there are classes which are not iterative sets
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 1.3)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.19
A Reaction
In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20.
The
15 ideas
with the same theme
[sets as a well-founded hierarchy built from scratch]:
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10484
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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]
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17801
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The set hierarchy doesn't rely on the dubious notion of 'generating' them
[Mayberry]
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10565
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There is no stage at which we can take all the sets to have been generated
[Fine,K]
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13019
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The Iterative Conception says everything appears at a stage, derived from the preceding appearances
[Maddy]
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13640
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Russell's paradox shows that there are classes which are not iterative sets
[Shapiro]
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13654
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It is central to the iterative conception that membership is well-founded, with no infinite descending chains
[Shapiro]
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13666
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Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
[Shapiro]
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9617
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The 'iterative' view says sets start with the empty set and build up
[Brown,JR]
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10405
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In the iterative conception of sets, they form a natural hierarchy
[Swoyer]
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10708
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Nowadays we derive our conception of collections from the dependence between them
[Potter]
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15900
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The iterative conception of set wasn't suggested until 1947
[Lavine]
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15931
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The iterative conception needs the Axiom of Infinity, to show how far we can iterate
[Lavine]
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15932
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The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs
[Lavine]
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23624
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The iterative conception has to appropriate Replacement, to justify the ordinals
[Hossack]
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