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Like the well-ordering principle, Zorn's Lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms of set theory is sufficient to prove the other. It is probably the most useful of all equivalents of the axiom of choice and occurs in the proofs of several theorems of crucial importance, for instance the Hahn-Banach theorem in functional analysis, the theorem that every vector space has a basis, the theorem in abstract algebra that every commutative ring has a [prime ideal]? and the theorem that every field has an [algebraic closure]?. |
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Like the well-ordering principle, Zorn's Lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms of set theory is sufficient to prove the other. It is probably the most useful of all equivalents of the axiom of choice and occurs in the proofs of several theorems of crucial importance, for instance the Hahn-Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every commutative ring has a [prime ideal]? and that every field has an [algebraic closure]?. |
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A sketch of the proof follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and every element has a bigger one. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually define the function b, we need to employ the axiom of choice. |
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A sketch of the proof of Zorn's lemma follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and every element has a bigger one. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually define the function b, we need to employ the axiom of choice. |
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