Cantorian Set Theory and Limitation of Size
Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.
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The Cantorian origins of set theory
Cantors theory of infinity
The ordinal theory of powers
Cantors theory of number
The origin of the limitation of size idea
The limitation of size argument and axiomatic set theory
The completability of sets
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1ndeed absolute collections abstraction actual infinite aleph antinomies argument assumed assumption attempt axiom of choice axiom of infinity axiom of replacement axiom of separation axiomatic set theory Borel Burali-Forti Cantor's theory Cantorian certainly clear completed comprehension construction continuum hypothesis continuum problem contradiction countable counting crucial Dedekind defined definition doctrine domain elements enumeral example explain extension finite numbers follows Fraenkel Frege function Godel Hausdorff Hessenberg heuristic hierarchy idea important impredicative infinite sets infinity intuition Jourdain Kant Kant's large cardinal later limitation limitation of size logical Mirimanoff natural numbers Neumann notion number sequence number-class order-type ordinal numbers ordinal theory paradoxes point-sets possible power-set axiom precisely principle proof properties proved real numbers reductionism reductionist remarks Russell Russell's seems sense set theory set-theoretic Shoenfield simple stage subset suggests theory of cardinality theory of number tion transfinite numbers transfinite ordinal uncountable universe well-ordered set well-ordering theorem
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