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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.
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.
There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice (AC). This is incorrect. Transfinite induction can be applied to any wellordered set. However, frequently proofs or constructions using transfinite induction also use the axiom of choice to wellorder a set. ..." is changed to
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.
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:
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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.