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The Lumenary
九至十二年级Foundations计划中

Russell's paradox and why sets need axioms

Naive comprehension, the idea that every property defines a set, is inconsistent: the set of all sets that are not members of themselves cannot exist. Axiomatic set theory replaces comprehension with safer principles such as separation.

math.found.russell-paradox

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前置知识

Set-builder notation, quantifiers and infinite setsProof: direct, contrapositive, contradiction and induction

10 项必需的前置知识（一直追溯到底）

你将学会

  • State the naive comprehension principle and derive Russell's contradiction from it.
  • Explain how the separation axiom, which only forms {x ∈ A : P(x)} inside an existing set A, blocks the paradox, and deduce that there is no set of all sets.
  • Distinguish naive set theory, a useful informal language, from axiomatic set theory, a precise theory.

定义

  • Naive comprehension: for every property P there is a set {x : P(x)}.
  • Russell's set is R = {x : x ∉ x}.
  • Separation: for every set A and property P, {x ∈ A : P(x)} is a set.

定理

  • Naive comprehension is inconsistent: it gives R ∈ R if and only if R ∉ R.
  • With separation, for any set A the set {x ∈ A : x ∉ x} is not an element of A; hence no set contains every set.
  • Ordinary mathematics can be carried out in axiomatic set theories such as ZFC that avoid the paradox.

常见误解

  • Thinking the paradox shows mathematics is inconsistent; it shows naive comprehension is.
  • Treating Russell's paradox and the liar paradox as the same argument; they are analogous, not identical.
  • Believing ZF avoids the paradox by banning x ∈ x. ZF does prove x ∉ x, but from the separate axiom of foundation; the paradox is blocked by restricting comprehension to separation.

可选发现

延伸阅读

  • Paul R. Halmos, Naive Set Theory (1960)
  • Herbert B. Enderton, Elements of Set Theory (1977)