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Proofs or constructions using induction and recursion often use the axiom of choice to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice. [4]
Frequently, the axiom of choice allows generalizing a theorem to "larger" objects. For example, it is provable without the axiom of choice that every vector space of finite dimension has a basis, but the generalization to all vector spaces requires the axiom of choice.
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion.
The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
Axiom of choice. Axiom of countable choice; Axiom of dependent choice; Zorn's lemma; Axiom of power set; Boolean-valued model; Burali-Forti paradox; Cantor's back-and-forth method; Cantor's diagonal argument; Cantor's first uncountability proof; Cantor's paradox; Cantor's theorem; Cantor–Bernstein–Schroeder theorem; Cardinal number. Aleph ...
Transfinite induction has nothing to do, formally speaking, with the axiom of choice. In practice, though, the arguments for which you want to use transfinite induction, will generally require AC. In particular, one frequently wants to exhaust all the elements of some set, one at a time, in an inductive process.
This form of induction, when applied to a set of ordinal numbers (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields. Proofs by transfinite induction typically distinguish three cases:
A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the axiom of dependent choice). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which ...