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).

Formal statement

The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (x_n) in X such that x_nR‌x_n+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 DC_R.

Use

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

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 is also equivalent over ZF to the Löwenheim–Skolem theorem.

Relation with other axioms

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. This follows because the Solovay model satisfies ZF + DC and every set of reals 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.

Footnotes

1. ↑ 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

• 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.

Set theory
Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union
Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram
Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem
Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine