Editing Zorn's lemma (section)

==Equivalent forms of Zorn's lemma==
{{See also|Axiom of choice#Equivalents}}

Zorn's lemma is equivalent (in [[Zermelo–Fraenkel set theory|ZF]]) to three main results:
# [[Hausdorff maximal principle]]
# [[Axiom of choice]]
# [[Well-ordering theorem]].

A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to [[Jerry Bona]]:
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"<ref>{{citation
 | last = Krantz | first = Steven G. | author-link = Steven G. Krantz
 | doi = 10.1007/978-1-4612-0115-1_9
 | pages = 121–126
 | publisher = Springer
 | title = Handbook of Logic and Proof Techniques for Computer Science
 | year = 2002| chapter = The Axiom of Choice | isbn = 978-1-4612-6619-8 }}.</ref>

Zorn's lemma is also equivalent to the [[strong completeness theorem]] of first-order logic.<ref>J.L. Bell & A.B. Slomson (1969). ''Models and Ultraproducts''. North Holland Publishing Company. Chapter 5, Theorem 4.3, page 103.</ref>

Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
# [[Banach's extension theorem]] which is used to prove one of the most fundamental results in functional analysis, the [[Hahn–Banach theorem]]
# Every vector space has a [[Basis (linear algebra)|basis]], a result from linear algebra (to which it is equivalent<ref>{{Cite book
|last = Blass
|first = Andreas
|title = Axiomatic Set Theory
|year = 1984
|chapter = Existence of bases implies the Axiom of Choice
|volume = 31
|pages = 31–33
|ref=blass
|doi=10.1090/conm/031/763890
|series = Contemporary Mathematics
|isbn = 9780821850268
}}</ref>). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis.
# Every commutative unital ring has a [[maximal ideal]], a result from ring theory known as [[Krull's theorem]], to which Zorn's lemma is equivalent<ref>{{cite journal|last=Hodges|first=W.|year=1979|title=Krull implies Zorn|journal=[[Journal of the London Mathematical Society]]|volume=s2-19|issue=2|pages=285–287|doi=10.1112/jlms/s2-19.2.285}}</ref>
# [[Tychonoff's theorem]] in topology (to which it is also equivalent<ref>{{cite journal
|last = Kelley
|first = John L.
|year = 1950
|title= The Tychonoff product theorem implies the axiom of choice
| journal= Fundamenta Mathematicae
| volume = 37
| pages = 75–76
|doi = 10.4064/fm-37-1-75-76
| ref=kelley
|doi-access = free
}}</ref>)
# Every [[proper filter]] is contained in an [[ultrafilter]], a result that yields the [[completeness theorem]] of [[first-order logic]]<ref>J.L. Bell & A.B. Slomson (1969). ''Models and Ultraproducts''. North Holland Publishing Company.</ref>

In this sense, Zorn's lemma is a powerful tool, applicable to many areas of mathematics.

=== Analogs under weakenings of the axiom of choice ===
{{See also|Axiom of dependent choice}}

A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the [[axiom of dependent choice]]). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element.<ref name="Wolk 1983">{{citation|title=On the principle of dependent choices and some forms of Zorn's lemma|first=Elliot S.|last=Wolk |journal=[[Canadian Mathematical Bulletin]]|volume=26|issue=3|pages=365–367 |year=1983 | doi=10.4153/CMB-1983-062-5 | doi-access=free }}</ref>

More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities.<ref name="Wolk 1983"/> In the limit where we allow arbitrarily large ordinals, we recover the proof of the full Zorn's lemma using the axiom of choice in the preceding section.