Hollis Frampton - Zorn's Lemma (1970)
Wednesday, February 20, 2008
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Zorn's Lemma stands for - Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.
It is named after the mathematician Max Zorn.
The terms are defined as follows. Suppose (P,≤) is the partially ordered set. A subset T is totally ordered if for any s, t ∈ T we have either s ≤ t or t ≤ s. Such a set T has an upper bound u ∈ P if t ≤ u for all t ∈ T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m ∈ P such that the only element x ∈ P with m ≥ x is x = m itself.
Like the well-ordering theorem, 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 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 ring has a maximal ideal and that every field has an algebraic closure.
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Zorn's Lemma stands for - Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.
It is named after the mathematician Max Zorn.
The terms are defined as follows. Suppose (P,≤) is the partially ordered set. A subset T is totally ordered if for any s, t ∈ T we have either s ≤ t or t ≤ s. Such a set T has an upper bound u ∈ P if t ≤ u for all t ∈ T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m ∈ P such that the only element x ∈ P with m ≥ x is x = m itself.
Like the well-ordering theorem, 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 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 ring has a maximal ideal and that every field has an algebraic closure.
"Zorns Lemma is a major poetic work. Created and put together by a very clear eye head, this original and complex abstract work moves beyond the letters of the alphabet, beyond words and beyond Freud. If you don't understand it the first time you see it, don't despair, see it again! When you finally 'get it,' a small light, possibly a candle, will light itself inside your forehead."
-- Ernie Gehr.
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