Axiom of dependent choice
In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop most of real analysis. It was introduced by Bernays (1942).
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Formal statement[edit]
The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (xn) in X such that xnRxn+1 for each n in N. (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way.
If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DCR.
Use[edit]
DC is the fragment of AC 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]
DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree with ω levels has a branch.
It is also equivalent[1] to the Baire category theorem for complete metric spaces.
DC also is (over the theory ZF) equivalent to the statement "Lowenheim-Skolem Theorem" - Statement: 'if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.'
Relation with other axioms[edit]
Unlike full AC, DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.
The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.
Footnotes[edit]
- ^ 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.
References[edit]
- Bernays, Paul (1942), "A system of axiomatic set theory. III. Infinity and enumerability. Analysis.", J. Symbolic Logic, 7: 65–89, MR 0006333
- Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
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