Axiom of dependent choice

Summary

In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice () that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.[a]

Formal statementEdit

A homogeneous relation   on   is called a total relation if for every   there exists some   such that   is true.

The axiom of dependent choice can be stated as follows: For every nonempty set   and every total relation   on   there exists a sequence   in   such that

  for all  

If the set   above is restricted to be the set of all real numbers, then the resulting axiom is denoted by  

UseEdit

Even without such an axiom, for any  , one can use ordinary mathematical induction to form the first   terms of such a sequence. The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way.

The axiom   is the fragment of   that is required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of those choices cannot be made independently of previous choices.

Equivalent statementsEdit

Over Zermelo–Fraenkel set theory  ,   is equivalent to the Baire category theorem for complete metric spaces.[1]

It is also equivalent over   to the Löwenheim–Skolem theorem.[b][2]

  is also equivalent over   to the statement that every pruned tree with   levels has a branch (proof below).

Furthermore,   is equivalent to a weakened form of Zorn's lemma; specifically   is equivalent to the statement that any partial order such that every well-ordered chain is finite and bounded, must have a maximal element.[3]

Relation with other axiomsEdit

Unlike full  ,   is insufficient to prove (given  ) that there is a non-measurable set of real numbers, or that there is a set of real numbers without the property of Baire or without the perfect set property. This follows because the Solovay model satisfies  , and every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property.

The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]

NotesEdit

  1. ^ "The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." Bernays, Paul (1942). "Part III. Infinity and enumerability. Analysis" (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 7 (2): 65–89. doi:10.2307/2266303. JSTOR 2266303. MR 0006333. The axiom of dependent choice is stated on p. 86.
  2. ^ Moore states that "Principle of Dependent Choices   Löwenheim–Skolem theorem" — that is,   implies the Löwenheim–Skolem theorem. See table Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its origins, development, and influence. Springer. p. 325. ISBN 0-387-90670-3.

ReferencesEdit

  1. ^ "The Baire category theorem implies the principle of dependent choices." Blair, Charles E. (1977). "The Baire category theorem implies the principle of dependent choices". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 25 (10): 933–934.
  2. ^ The converse is proved in Boolos, George S.; Jeffrey, Richard C. (1989). Computability and Logic (3rd ed.). Cambridge University Press. pp. 155–156. ISBN 0-521-38026-X.
  3. ^ Wolk, Elliot S. (1983), On the principle of dependent choices and some forms of Zorn's lemma, vol. 26, Canadian Mathematical Bulletin, pp. 365–367, doi:10.4153/CMB-1983-062-5
  4. ^ Bernays proved that the axiom of dependent choice implies the axiom of countable choice See esp. p. 86 in Bernays, Paul (1942). "Part III. Infinity and enumerability. Analysis" (PDF). Journal of Symbolic Logic. A system of axiomatic set theory. 7 (2): 65–89. doi:10.2307/2266303. JSTOR 2266303. MR 0006333.
  5. ^ For a proof that the Axiom of Countable Choice does not imply the Axiom of Dependent Choice see Jech, Thomas (1973), The Axiom of Choice, North Holland, pp. 130–131, ISBN 978-0-486-46624-8