The axiom of choice and its equivalents

इस पर आधारित

20 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

• 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).

परिभाषा

• 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.

प्रमेय

• 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.

आम भ्रांतियाँ

• 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.

आगे ले जाता है

वैकल्पिक खोजें

• 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).

आगे पढ़ने के लिए

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