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The Lumenary
本科高年级Foundations计划中

The axiom of choice and its equivalents

The axiom of choice, its equivalents Zorn's lemma and the well-ordering theorem, typical applications in algebra and analysis, and its non-constructive consequences.

math.found.axiom-of-choice

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前置知识

Ordinals and cardinals

20 项必需的前置知识（一直追溯到底）

你将学会

  • Prove, over ZF, that the axiom of choice, Zorn's lemma and the well-ordering theorem are equivalent.
  • Use Zorn's lemma to prove that every vector space has a basis and that every non-zero commutative ring with identity has a maximal ideal.
  • Describe results, such as the existence of a non-measurable set of reals, that are provable with choice but not in ZF alone (relative to the consistency of an inaccessible cardinal).

定义

  • Axiom of choice (AC): every family of non-empty sets has a choice function.
  • Well-ordering theorem: every set admits a well-order.
  • Countable choice and dependent choice (DC) are weaker principles that suffice for much of classical analysis.

定理

  • Over ZF, AC, Zorn's lemma and the well-ordering theorem are equivalent.
  • Gödel: if ZF is consistent, so is ZFC. Cohen (1963): if ZF is consistent, so is ZF + ¬AC.
  • AC implies that some subset of ℝ is not Lebesgue measurable (Vitali). Solovay showed that, if an inaccessible cardinal is consistent with ZFC, then ZF + DC is consistent with every set of reals being Lebesgue measurable.

常见误解

  • Thinking choice is only needed for uncountable families; choosing from countably many two-element sets can already require it.
  • Believing the Banach–Tarski paradox refutes AC; it shows that some sets cannot be assigned a volume invariant under rigid motions.
  • Assuming every proof that uses Zorn's lemma can easily be made constructive.

通向

Axiomatic set theory: hierarchy, cardinal arithmetic and L

可选发现

  • Measure theory and integration Vitali's construction of a non-measurable set uses AC; this is one reason Lebesgue measure is defined on a σ-algebra of sets rather than on all subsets of ℝ.
  • Functional analysis The Hahn–Banach theorem is usually proved with Zorn's lemma, although it follows from the strictly weaker ultrafilter lemma.
  • Advanced linear algebra Over ZF, 'every vector space has a basis' is equivalent to the axiom of choice (Blass, 1984).

延伸阅读

  • Thomas Jech, The Axiom of Choice (1973)
  • Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (1980)