Axiom of dependent choice
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.
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 .
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.
|Proof that Every pruned tree with ω levels has a branch|
| Let be an entire binary relation on . The strategy is to define a tree on of finite sequences whose neighboring elements satisfy Then a branch through is an infinite sequence whose neighboring elements satisfy Start by defining if for Since is entire, is a pruned tree with levels. Thus, has a branch So, for all which implies Therefore, is true.
Let be a pruned tree on with levels. The strategy is to define a binary relation on so that produces a sequence where and is a strictly increasing function. Then the infinite sequence is a branch. (This proof only needs to prove this for ) Start by defining if is an initial subsequence of and Since is a pruned tree with levels, is entire. Therefore, implies that there is an infinite sequence such that Now for some Let be the last element of Then For all the sequence belongs to because it is an initial subsequence of or it is a Therefore, is a branch.
Relation with other axioms
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.
- 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.
- 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.
- 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.
- 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.
- Bernays, Paul (1942), "A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis.", Journal of Symbolic Logic, 7: 65–89, doi:10.2307/2266303, JSTOR 2266303, MR 0006333.
- Boolos, George S.; Jeffrey, Richard C. (1989), Computability and Logic (3rd ed.), Cambridge University Press, ISBN 0-521-38026-X.
- Jech, Thomas (1973), The Axiom of Choice, North Holland, ISBN 978-0-486-46624-8.
- Jech, Thomas (2003), Set theory (Third Millennium ed.), Springer-Verlag, ISBN 3-540-44085-2, OCLC 174929965, Zbl 1007.03002.
- Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Springer, ISBN 0-387-90670-3.