Inverse operations with transfinite numbers and the Kalam cosmological argument (1995)

In "Reply To Smith: On The Finitude Of The Past" [1], Professor William Craig writes: I reiterate that Smith has yet to deal with my strongest arguments in favour of the impossibility of the existence of an actual infinite, those based on inverse operations performed with transfinite numbers. [2] I think that this claim is mistaken; for: (i) there is no problem about allowing the inverse operations in question--subtraction, division, extracting roots, etc.--into transfinite ordinal arithmetic[3]; and (ii) there is no problem about the exclusion of these operations from transfinite cardinal arithmetic. I shall take up these points in turn.
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