Full Idea
Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal.
Gist of Idea
Transfinite induction moves from all cases, up to the limit ordinal
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11)
Book Reference
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.83