One example is the axiom of dependent choice (DC).

An axiom of dependent choice, which is much weaker than the usual axiom of choice.

For inductions and recursions of countable length, the weaker axiom of dependent choice is sufficient.

Axiom of dependent choice.)

The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.

Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).

The principles of bar induction and bar recursion are the intuitionistic equivalents of the axiom of dependent choices.

ZF stands for Zermelo-Fraenkel set theory, and DC for the axiom of dependent choice.

Furthermore, this is possible whilst assuming the Axiom of dependent choice, which is weaker than AC but sufficient to develop most of real analysis.

It has been shown that neither ZF theory nor ZF theory with the axiom of dependent choice is sufficient.

König's lemma may be considered to be a choice principle; the first proof above illustrates the relationship between the lemma and the axiom of dependent choice.

Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.

Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.

However, L(R) will still satisfy the axiom of dependent choice, given only that the von Neumann universe, V, also satisfies that axiom.

The axiom, which is to be understood in the context of ZF plus DC (the axiom of dependent choice for real numbers), states two things:

AC clearly implies the axiom of dependent choice (DC), and DC is sufficient to show AC.

Given the axiom of dependent choice, this is equivalent to just saying that the set is totally ordered and there is no infinite decreasing sequence, something perhaps easier to visualize.

König's lemma is essentially the restriction of the axiom of dependent choice to entire relations R such that for each x there are only finitely many z such that xRz.

In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice (AC) which is still sufficient to develop most of real analysis.

Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.

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 choices merely says that we can form a whole sequence this way.

Thus, by the axiom of dependent choice, there is some sequence (a) in S satisfying aRa for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.

Because there are models of Zermelo-Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.