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Single Idea 13015
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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Full Idea
Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list.
Gist of Idea
Zermelo used Foundation to block paradox, but then decided that only Separation was needed
Source
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2
Book Ref
-: 'Journal of Symbolic Logic' [-], p.484
A Reaction
Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity.
The
14 ideas
from 'Investigations in the Foundations of Set Theory I'
13017
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Zermelo introduced Pairing in 1930, and it seems fairly obvious
[Zermelo, by Maddy]
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13015
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Zermelo used Foundation to block paradox, but then decided that only Separation was needed
[Zermelo, by Maddy]
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13486
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Not every predicate has an extension, but Separation picks the members that satisfy a predicate
[Zermelo, by Hart,WD]
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13020
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The Axiom of Separation requires set generation up to one step back from contradiction
[Zermelo, by Maddy]
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13487
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In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals
[Zermelo, by Hart,WD]
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18178
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For Zermelo the successor of n is {n} (rather than n U {n})
[Zermelo, by Maddy]
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13027
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Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets
[Zermelo, by Maddy]
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9627
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Different versions of set theory result in different underlying structures for numbers
[Zermelo, by Brown,JR]
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15924
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Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them?
[Zermelo, by Lavine]
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10870
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ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice
[Zermelo, by Clegg]
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13012
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Zermelo published his axioms in 1908, to secure a controversial proof
[Zermelo, by Maddy]
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17609
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Set theory can be reduced to a few definitions and seven independent axioms
[Zermelo]
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17607
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Set theory investigates number, order and function, showing logical foundations for mathematics
[Zermelo]
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17608
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We take set theory as given, and retain everything valuable, while avoiding contradictions
[Zermelo]
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