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Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
The proof is by transfinite induction. Let be a limit ordinal (the induction is trivial for successor ordinals), and for each <, let {} be a partition of satisfying the requirements of the theorem.
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.
Gentzen's theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order Peano arithmetic (PA) but does not contain PA. For example, it does not prove ordinary mathematical induction for all formulae, whereas PA does (since all instances of induction are axioms of PA).
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:
The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction. The values of -tuples of parameters are well-ordered by the product ordering. The formulas with parameters are well-ordered by the ordered sum (by Gödel numbers) of well-orderings.
Transfinite induction works for the entire ordered set. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice). Every subordering is isomorphic to an initial segment.
Transfinite induction, an extension of mathematical induction to well-ordered sets Transfinite recursion; Transfinite arithmetic, the generalization of elementary arithmetic to infinite quantities; Transfinite interpolation, a method in numerical analysis to construct functions over a planar domain so that they match a given function on the ...