체르멜로-프란켈 집합론

수학노트
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노트

  • Most notable is the Continuum Hypothesis discussed on page 1127, which was proved independent of ZFC by Paul Cohen in 1963.[1]
  • The standard axioms of set theory (ZFC, the Zermelo-Fraenkel axioms with the Axiom of Choice) give us only partial information here.[2]
  • As mentioned above ZFC requires an infinite number of first-order axioms.[3]
  • Everything in standard ZFC ZFC is a pure set, which we will call simply a set; but there are also variations with urelements and classes.[4]
  • ZFC ZFC adds (9) and is thus the strongest version without classes or additional axioms.[4]
  • One often adds axioms for large cardinals to ZFC ZFC .[4]
  • Adding this axiom to ZFC ZFC makes Tarski–Grothendieck set theory ( TG TG ).[4]
  • If the axiom of choice is added it is known as ZFC.[5]
  • ZFC is the acronym for Zermelo–Fraenkel set theory with the axiom of choice, formulated in first-order logic.[6]
  • All formulations of ZFC imply that at least one set exists.[7]
  • First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty.[7]
  • Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC.[7]
  • Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent).[7]
  • ZFC is an axiom system formulated in first-order logic with equality and with only one binary relation symbol \(\in\) for membership.[8]
  • In ZFC one can develop the Cantorian theory of transfinite (i.e., infinite) ordinal and cardinal numbers.[8]
  • In ZFC, one identifies the finite ordinals with the natural numbers.[8]
  • In the case of exponentiation of singular cardinals, ZFC has a lot more to say.[8]

소스

  1. ↑ Note (c) for Implications for Mathematics and Its Foundations: A New Kind of Science
  2. ↑ Set Theory
  3. ↑ ZermeloFraenkelSetTheory
  4. ↑ 4.0 4.1 4.2 4.3 ZFC in nLab
  5. ↑ Zermelo-Fraenkel set theory
  6. ↑ Encyclopedia of Mathematics
  7. ↑ 7.0 7.1 7.2 7.3 Zermelo–Fraenkel set theory
  8. ↑ 8.0 8.1 8.2 8.3 Set Theory (Stanford Encyclopedia of Philosophy)

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'zermelo'}, {'OP': '*'}, {'LOWER': 'fraenkel'}, {'LOWER': 'set'}, {'LEMMA': 'theory'}]
  • [{'LEMMA': 'ZFC'}]
  • [{'LOWER': 'zf'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'zfc'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'set'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'zf'}, {'LOWER': 'set'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'zfc'}, {'LOWER': 'set'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'zermelo'}, {'OP': '*'}, {'LOWER': 'fraenkel'}, {'LOWER': 'set'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'zermelo'}, {'OP': '*'}, {'LOWER': 'fraenkel'}, {'LEMMA': 'axiom'}]
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