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 some such that is true.) 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.
It is also equivalent over to the Löwenheim–Skolem theorem.
|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.
The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.
- "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.
- 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.
- "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. Astronom. Phys. 25 (10): 933–934.
- 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.
- 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.
- 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
- Jech, Thomas (2003). Set Theory (Third Millennium ed.). Springer-Verlag. ISBN 3-540-44085-2. OCLC 174929965. Zbl 1007.03002.