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Does the principle of mathematical induction extend to a cardinality larger than that of the countably infinite?

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up vote 8 down vote accepted

Mathematical induction can be extended, one of the usual ways to extend it is transfinite induction, as mentioned by PEV.

A more general way requires slightly more terms:

Let $R$ be a partial order. We say that $R$ is well-founded if for every non-empty class $A$ there is an $R$-minimal element in $A$. We say that $R$ is set-like if for every $x$ the class $\{y | yRx\}$ is a set.

Suppose that $A$ is a class, $R$ is a well-founded partial order which is set like (at least on the elements of $A$), then if $B\subseteq A$ such that for every $x\in A$ we have $(yRx \rightarrow y\in B) \rightarrow x\in B$ then $A=B$.

This is the same way induction is defined on the natural numbers, even if it isn't always clear how from this clumsy definition one can prove things by induction as usually.

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I think the R doesn't have to be set-like. – D.F.F Dec 9 '15 at 19:04

Yes see this (site about transfinite induction).

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