Editing Well-ordering theorem (section)

==History==
[[Georg Cantor]] considered the well-ordering theorem to be a "fundamental principle of thought".<ref>Georg Cantor (1883), “Ueber unendliche, lineare Punktmannichfaltigkeiten”, ''Mathematische Annalen'' 21, pp. 545–591.</ref>  However, it is considered difficult or even impossible to visualize a well-ordering of <math>\mathbb{R}</math>, the set of all [[real number]]s; such a visualization would have to incorporate the axiom of choice.<ref>{{cite book |url=https://books.google.com/books?id=RXzsAwAAQBAJ&pg=PA174 |title=The Logic of Infinity |page=174 |publisher=Cambridge University Press |isbn=978-1-1070-5831-6 |first=Barnaby |last=Sheppard |year=2014 }}</ref> In 1904, [[Gyula Kőnig]] claimed to have proven that such a well-ordering cannot exist.  A few weeks later, [[Felix Hausdorff]] found a mistake in the proof.<ref>{{citation|title=Hausdorff on Ordered Sets|volume=25|series=History of Mathematics|first=J. M.|last=Plotkin|publisher=American Mathematical Society|isbn=9780821890516|year=2005|contribution=Introduction to "The Concept of Power in Set Theory"|pages=23–30|url=https://books.google.com/books?id=M_skkA3r-QAC&pg=PA23}}</ref>  It turned out, though, that in [[first-order logic]] the well-ordering theorem is equivalent to the axiom of choice, in the sense that the [[Zermelo–Fraenkel axioms]] with the axiom of choice included are sufficient to prove the well-ordering theorem, and conversely, the Zermelo–Fraenkel axioms without the axiom of choice but with the well-ordering theorem included are sufficient to prove the axiom of choice. (The same applies to [[Zorn's lemma]].) In [[second-order logic]], however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.<ref>{{cite book |authorlink=Stewart Shapiro |first=Stewart |last=Shapiro |year=1991 |title=Foundations Without Foundationalism: A Case for Second-Order Logic |location=New York |publisher=Oxford University Press |isbn=0-19-853391-8 }}</ref>
There is a well-known joke about the three statements, and their relative amenability to intuition:<blockquote>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.|chapter=The Axiom of Choice|date=2002|pages=121–126|editor-last=Krantz|editor-first=Steven G.|publisher=Birkhäuser Boston|language=en|doi=10.1007/978-1-4612-0115-1_9|isbn=9781461201151|title=Handbook of Logic and Proof Techniques for Computer Science}}</ref></blockquote>