Axiom of dependent choice

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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.[1]

Formal statement[edit]

The axiom can be stated as follows: For every nonempty set and every entire binary relation on , there exists a sequence in such that for all . (Here, an entire binary relation on is one where for every , there exists a such that .) Note that even without such an axiom, one can use ordinary mathematical induction to form the first terms of such a sequence, for every ; the axiom of dependent choice says that we can form a whole sequence this way.

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

Use[edit]

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 statements[edit]

is equivalent (over Zermelo–Fraenkel set theory ) to the statement that every pruned tree with levels has a branch.

is also equivalent over to the Baire category theorem for complete metric spaces.[2]

It is also equivalent over to the Löwenheim–Skolem theorem.[3]

Relation with other axioms[edit]

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]

Footnotes[edit]

  1. ^ Bernays starts his article by stating: "The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." (Bernays 1942, p. 65). Bernays introduces the axiom of dependent choice on page 86.
  2. ^ Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934.
  3. ^ Moore 1982, p. 325. Moore's table states that "Principle of Dependent Choices" "Löwenheim–Skolem theorem"—that is, implies the Löwenheim–Skolem theorem. The converse is proved in Boolos and Jeffrey 1989, pp. 155–156.
  4. ^ Bernays proved that the axiom of dependent choice implies the axiom of countable choice (Bernays 1942, p. 86). For a proof that the axiom of countable choice does not imply the axiom of dependent choice, see Jech 1973, pp. 130–131.

References[edit]