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Single Idea 13012
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
]
Full Idea
Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'.
Gist of Idea
Zermelo published his axioms in 1908, to secure a controversial proof
Source
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1
Book Ref
-: 'Journal of Symbolic Logic' [-], p.483
A Reaction
This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident.
The
14 ideas
from 'Investigations in the Foundations of Set Theory I'
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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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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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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17608
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We take set theory as given, and retain everything valuable, while avoiding contradictions
[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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