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facenian1
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Is transfinite induction a theorem o an axiom?
Transfinite induction is a mathematical proof technique used to show that a statement holds for all infinite objects in a particular set.
The main difference between transfinite induction and regular induction is that transfinite induction is used to prove statements about infinite sets, while regular induction is used to prove statements about finite sets.
Transfinite induction is both a theorem and an axiom. It is an axiom in set theory, meaning that it is assumed to be true without proof. However, it can also be proven as a theorem within the framework of set theory.
Transfinite induction allows mathematicians to prove statements about infinite sets in a rigorous and systematic way. It also allows for the extension of regular induction to infinite sets.
Transfinite induction can only be applied to well-ordered sets, which have a clear notion of "first" and "next" elements. This means that it cannot be used to prove statements about all infinite sets, only those that can be well-ordered.