- Zorn's lemma
**Zorn's lemma**, also known as the**Kuratowski-Zorn lemma**, is a proposition ofset theory that states:Every

partially ordered set in which every chain (i.e. totally orderedsubset ) has anupper bound contains at least onemaximal element .It is named after the mathematician Max Zorn.

The terms are defined as follows. Suppose ("P",≤) is a

partially ordered set . A subset "T" is "totally ordered" if for any "s", "t" in "T" we have either "s" ≤ "t" or "t" ≤ "s". Such a set "T" has an "upper bound" "u" in "P" if "t" ≤ "u" for all "t" in "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" in "P" such that for no element "x" in "P", "m" < "x".Zorn's lemma is equivalent to the

well-ordering theorem and theaxiom of choice , in the sense that any one of them, together with theZermelo-Fraenkel axioms ofset theory , is sufficient to prove the others. It occurs in the proofs of several theorems of crucial importance, for instance theHahn-Banach theorem infunctional analysis , the theorem that everyvector space has a basis,Tychonoff's theorem intopology stating that every product ofcompact space s is compact, and the theorems inabstract algebra that every ring has amaximal ideal and that every field has analgebraic closure .**An example application**We will go over a typical application of Zorn's lemma: the proof that every ring "R" with unity contains a

maximal ideal . The set "P" here consists of all (two-sided) ideals in "R" except "R" itself, which is not empty since it contains at least the trivial ideal {0}. This set is partially ordered by set inclusion. We are done if we can find a maximal element in "P". The ideal "R" was excluded because maximal ideals by definition are not equal to "R".We want to apply Zorn's lemma, and so we take a totally ordered subset "T" of "P" and have to show that "T" has an upper bound, i.e. that there exists an ideal "I" ⊆ "R" which is bigger than all members of "T" but still smaller than "R" (otherwise it would not be in "P"). We take "I" to be the union of all the ideals in "T". "I" is an ideal: if "a" and "b" are elements of "I", then there exist two ideals "J", "K" ∈ "T" such that "a" is an element of "J" and "b" is an element of "K". Since "T" is totally ordered, we know that "J" ⊆ "K" or "K" ⊆ "J". In the first case, both "a" and "b" are members of the ideal "K", therefore their sum "a" + "b" is a member of "K", which shows that "a" + "b" is a member of "I". In the second case, both "a" and "b" are members of the ideal "J", and we conclude similarly that "a" + "b" ∈ "I". Furthermore, if "r" ∈ "R", then "ar" and "ra" are elements of "J" and hence elements of "I". We have shown that "I" is an ideal in "R".

Now comes the heart of the proof: why is "I" smaller than "R"? The crucial observation is that an ideal is equal to "R"

if and only if it contains 1. (It is clear that if it is equal to "R", then it must contain 1; on the other hand, if it contains 1 and "r" is an arbitrary element of "R", then "r1" = "r" is an element of the ideal, and so the ideal is equal to "R".) So, if "I" were equal to "R", then it would contain 1, and that means one of the members of "T" would contain 1 and would thus be equal to "R" - but we explicitly excluded "R" from "P".The condition of Zorn's lemma has been checked, and we thus get a maximal element in "P", in other words a maximal ideal in "R".

Note that the proof depends on the fact that our ring "R" has a multiplicative unit 1. Without this, the proof wouldn't work and indeed the statement would be false.

**Sketch of the proof of Zorn's lemma (from the axiom of choice)**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.

Using the function "b", we are going to define elements "a"

_{0}< "a"_{1}< "a"_{2}< "a"_{3}< ... in "P". This sequence is**really long**: the indices are not just thenatural number s, but all ordinals. In fact, the sequence is too long for the set "P"; there are too many ordinals, more than there are elements in any set, and the set "P" will be exhausted before long and then we will run into the desired contradiction.The "a

_{i}" are defined bytransfinite recursion : we pick "a"_{0}in "P" arbitrary (this is possible, since "P" contains an upper bound for the empty set and is thus not empty) and for any other ordinal "w" we set "a"_{"w"}= "b"({"a"_{"v"}: "v" < "w"}). Because the "a"_{"v"}are totally ordered, this is a well-founded definition.This proof shows that actually a slightly stronger version of Zorn's lemma is true::If "P" is a

poset in which everywell-order ed subset has an upper bound, and if "x" is any element of "P", then "P" has a maximal element that is greater than or equal to "x". That is, there is a maximal element which is comparable to "x".**History**The

Hausdorff maximal principle is an early statement similar to Zorn's lemma.K. Kuratowski proved in

1922 [*Casimir Kuratowski, "Une méthode d'élimination des nombres transfinis des raisonnements mathématiques", Fundamenta Mathematicae 3 (1922), pp. 76–108. [*] a version of Zorn's lemma close to its modern formulation (it applied to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by Max Zorn in*http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=3 icm*]1935 [*Max Zorn, "A remark on method in transfinite algebra", Bulletin of the American Mathematical Society 41 (1935), no. 10, pp. 667–670. doi|10.1090/S0002-9904-1935-06166-X*] , who proposed it as a newaxiom of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.The name "Zorn's lemma" appears to be due to

John Tukey , who used it in his book "Convergence and Uniformity in Topology" in 1940.Bourbaki 's "Théorie des Ensembles" of 1939 refers to a similar maximal principle as "le théorème de Zorn". [*Campbell, p. 82*]**References**

*"Set Theory for the Working Mathematician". Ciesielski, Krzysztof. Cambridge University Press, 1997. ISBN 0-521-59465-0

* cite journal

last = Campbell

first = Paul J.

year = 1978

month = February

title = The Origin of “Zorn's Lemma”

journal = Historia Mathematica

volume = 5

issue = 1

pages = 77–89

publisher = Elsevier

doi = 10.1016/0315-0860(78)90136-2**External links*** [

*http://www.apronus.com/provenmath/choice.htm Zorn's Lemma at ProvenMath*] contains a formal proof down to the finest detail of the equivalence of the axiom of choice and Zorn's Lemma.

* [*http://us.metamath.org/mpegif/zorn.html Zorn's Lemma*] atMetamath is another formal proof. ( [*http://us.metamath.org/mpeuni/zorn.html Unicode version*] for recent browsers.)

*Wikimedia Foundation.
2010.*

### См. также в других словарях:

**Zorn's lemma**— /zawrnz/, Math. a theorem of set theory that if every totally ordered subset of a nonempty partially ordered set has an upper bound, then there is an element in the set such that the set contains no element greater than the specified given… … Universalium**zorn's lemma**— ˈzȯ(ə)rnz , ˈtsȯ noun Usage: usually capitalized Z Etymology: after Max August Zorn died 1993 American (German born) mathematician : a lemma in set theory: if S is partially ordered and if each subset for which every pair of elements is related … Useful english dictionary**Zorn's lemma**— noun Etymology: Max August Zorn died 1993 German mathematician Date: circa 1950 a lemma in set theory: if a set S is partially ordered and if each subset for which every pair of elements is related by exactly one of the relationships “less than,” … New Collegiate Dictionary**Zorn's lemma**— A proposition in set theory equivalent to the axiom of choice . Call a set A a chain if for any two members B and C, either B is a subset of C or C is a subset of B. Now consider a set D with the properties that for every chain E that is a subset … Philosophy dictionary**Zorn's lemma**— noun A proposition of set theory stating that every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element … Wiktionary**Zorn's Lemma (film)**— Infobox Film name = Zorn s Lemma caption = director = Hollis Frampton producer = writer = starring = music = cinematography = editing = distributor = released = runtime = 60 min. country = flagicon|USA USA awards = language = English budget =… … Wikipedia**Zorn's Law**— * Zorn s law is a maxim coined by Chicago Tribune columnist Eric Zorn as a Wikipedia prank. * Zorn s lemma is a proposition used in many areas of theoretical mathematics … Wikipedia**Lemma (mathematics)**— In mathematics, a lemma (plural lemmata or lemmascite book |last= Higham |first= Nicholas J. |title= Handbook of Writing for the Mathematical Sciences |publisher= Society for Industrial and Applied Mathematics |year= 1998 |isbn= 0898714206 |pages … Wikipedia**Lemma von Zorn**— Das Lemma von Zorn, auch bekannt als Lemma von Kuratowski Zorn, ist ein Theorem der Mengenlehre, genauer gesagt, der Zermelo Fraenkel Mengenlehre, die das Auswahlaxiom einbezieht. Es ist benannt nach dem deutsch amerikanischen Mathematiker Max… … Deutsch Wikipedia**Lemma von Teichmüller-Tukey**— Das Lemma von Teichmüller Tukey (nach Oswald Teichmüller und John W. Tukey), manchmal auch nur Lemma von Tukey genannt, ist ein Satz aus der Mengenlehre. Es ist im Rahmen der Mengenlehre auf Grundlage der ZF Axiome äquivalent zum Auswahlaxiom und … Deutsch Wikipedia