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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]
Just to mention, I have seen the term "principle of transfinite induction" used in the literature to actually refer to forms of the Axiom of Choice such as Zorn's Lemma. For example, there is Terence Tao's classic text Analysis I. In Section 8.4 in this book, he uses the phrase "principle of transfinite induction" to refer to Zorn's Lemma.
2) I have no idea what do you mean by "transfinite recursion/induction available for all sets" either. Transfinite induction has nothing to do with the axiom of choice, whether you do it for ordinals or for well-founded relations. For example, ∈-induction applies to all sets, and it does not need the axiom of choice, only the axiom of foundation.
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 mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals , which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals , which are ordinal numbers used to provide an ordering ...
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.
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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.