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. A homogeneous relation on is called a total relation if for every there exists some such that is true. The axiom of dependent choice can be stated as follows: For every nonempty set and every total relation on there exists a sequence in such that for all In fact, x0 may be taken to be any desired element of X. (To see this, apply the axiom as stated above to the set of finite sequences that start with x0 and in which subsequent terms are in relation , together with the total relation on this set of the second sequence being obtained from the first by appending a single term.) If the set above is restricted to be the set of all real numbers, then the resulting axiom is denoted by Even without such an axiom, for any , one can use ordinary mathematical induction to form the first terms of such a sequence. The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way. The axiom 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. Over (Zermelo–Fraenkel set theory without the axiom of choice), is equivalent to the for complete metric spaces. It is also equivalent over to the Löwenheim–Skolem theorem. is also equivalent over to the statement that every pruned tree with levels has a branch (proof below). Furthermore, is equivalent to a weakened form of Zorn's lemma; specifically is equivalent to the statement that any partial order such that every well-ordered chain is finite and bounded, must have a maximal element.

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The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N. The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC).

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