Well-ordering theorem

In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).^[1]^[2] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.^[3] One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique.^[3] One famous consequence of the theorem is the Banach–Tarski paradox.

History

Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought".^[4] However, it is considered difficult or even impossible to visualize a well-ordering of $\mathbb {R}$ ; such a visualization would have to incorporate the axiom of choice.^[5] 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.^[6] 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.^[7]

There is a well-known joke about the three statements, and their relative amenability to intuition:

The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?^[8]

Proof from axiom of choice

The well-ordering theorem follows from the axiom of choice as follows.^[9]

Let the set we are trying to well-order be $A$ , and let $f$ be a choice function for the family of non-empty subsets of $A$ . For every ordinal $\alpha$ , define an element $a_{\alpha }$ that is in $A$ by setting $a_{\alpha }\ =\ f(A\smallsetminus \{a_{\xi }\mid \xi <\alpha \})$ if this complement $A\smallsetminus \{a_{\xi }\mid \xi <\alpha \}$ is nonempty, or leave $a_{\alpha }$ undefined if it is. That is, $a_{\alpha }$ is chosen from the set of elements of $A$ that have not yet been assigned a place in the ordering (or undefined if the entirety of $A$ has been successfully enumerated). Then the order $<$ on $A$ defined by $a_{\alpha }<a_{\beta }$ if and only if $\alpha <\beta$ (in the usual well-order of the ordinals) is a well-order of $A$ as desired, of order type $\sup\{\alpha \mid a_{\alpha }{\text{ is defined}}\}$ .

Proof of axiom of choice

The axiom of choice can be proven from the well-ordering theorem as follows.

To make a choice function for a collection of non-empty sets,

E

, take the union of the sets in

E

and call it

X

. There exists a well-ordering of

X

; let

R

be such an ordering. The function that to each set

S

of

E

associates the smallest element of

S

, as ordered by (the restriction to

S

of)

R

, is a choice function for the collection

E

.

An essential point of this proof is that it involves only a single arbitrary choice, that of $R$ ; applying the well-ordering theorem to each member $S$ of $E$ separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each $S$ a well-ordering would require just as many choices as simply choosing an element from each $S$ . Particularly, if $E$ contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of Zermelo-Fraenkel set theory without the axiom of choice.

Notes

^ Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Berlin: Springer. p. 14. ISBN 978-3-7643-8748-8.
^ Hazewinkel, Michiel (2001). Encyclopaedia of Mathematics: Supplement. Berlin: Springer. p. 458. ISBN 1-4020-0198-3.
^ ^a ^b Thierry, Vialar (1945). Handbook of Mathematics. Norderstedt: Springer. p. 23. ISBN 978-2-95-519901-5.
^ Georg Cantor (1883), “Ueber unendliche, lineare Punktmannichfaltigkeiten”, Mathematische Annalen 21, pp. 545–591.
^ Sheppard, Barnaby (2014). The Logic of Infinity. Cambridge University Press. p. 174. ISBN 978-1-1070-5831-6.
^ Plotkin, J. M. (2005), "Introduction to "The Concept of Power in Set Theory"", Hausdorff on Ordered Sets, History of Mathematics, vol. 25, American Mathematical Society, pp. 23–30, ISBN 9780821890516
^ Shapiro, Stewart (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. New York: Oxford University Press. ISBN 0-19-853391-8.
^ Krantz, Steven G. (2002), "The Axiom of Choice", in Krantz, Steven G. (ed.), Handbook of Logic and Proof Techniques for Computer Science, Birkhäuser Boston, pp. 121–126, doi:10.1007/978-1-4612-0115-1_9, ISBN 9781461201151
^ Jech, Thomas (2002). Set Theory (Third Millennium Edition). Springer. p. 48. ISBN 978-3-540-44085-7.

External links

Mizar system proof: http://mizar.org/version/current/html/wellord2.html

[1] Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Berlin: Springer. p. 14. ISBN 978-3-7643-8748-8.

[2] Hazewinkel, Michiel (2001). Encyclopaedia of Mathematics: Supplement. Berlin: Springer. p. 458. ISBN 1-4020-0198-3.

[zer-3] Thierry, Vialar (1945). Handbook of Mathematics. Norderstedt: Springer. p. 23. ISBN 978-2-95-519901-5.

[4] Georg Cantor (1883), “Ueber unendliche, lineare Punktmannichfaltigkeiten”, Mathematische Annalen 21, pp. 545–591.

[5] Sheppard, Barnaby (2014). The Logic of Infinity. Cambridge University Press. p. 174. ISBN 978-1-1070-5831-6.

[6] Plotkin, J. M. (2005), "Introduction to "The Concept of Power in Set Theory"", Hausdorff on Ordered Sets, History of Mathematics, vol. 25, American Mathematical Society, pp. 23–30, ISBN 9780821890516

[7] Shapiro, Stewart (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. New York: Oxford University Press. ISBN 0-19-853391-8.

[8] Krantz, Steven G. (2002), "The Axiom of Choice", in Krantz, Steven G. (ed.), Handbook of Logic and Proof Techniques for Computer Science, Birkhäuser Boston, pp. 121–126, doi:10.1007/978-1-4612-0115-1_9, ISBN 9781461201151

[9] Jech, Thomas (2002). Set Theory (Third Millennium Edition). Springer. p. 48. ISBN 978-3-540-44085-7.

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