Modifies a noun

• induction: His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction.

• number: Cantor held that these transfinite numbers had an actual existence, drawing on his early religious training to justify the assertion.

• ordinal: Our ability to think about transfinite ordinals is one of the things to be explained.

• cardinal: Recall that " addition " for transfinite cardinals is defined in terms of set union.

• mathematics: When the imaginary transfinite mathematics of the actual infinite is translated into real finite terms, the results are nonsensical.

• case: Finally, we extend the theory of needed reduction from the finite to the transfinite case.