1. I think Alex makes an inaccurate point here. From what I can tell, Craig mentions that Hart disagrees with Cantor's Intuition in footnote 33 of his Kalam Cosmological Argument (1979).
But why must we say that every class has a number? If the class of natural numbers is a potential infinite, increasing just as Thomson describes, then it is indefinite and cannot be said to possess an actually infinite number elements. See Hermann Weyl, 'Mathematics and Logic', American Mathematical Monthly 53 (1946): 2-13. That a potential infinite need not imply an actual infinite, as Cantor contended, is argued by Hart, 'Potential Infinite', pp. 254-64.Alex claims that Craig argues that Cantor's Thesis is refuted in W.D. Hart's ‘The Potential Infinite‘ (Proceedings of the Aristotelian Society, New Series, Vol. 76 (1975 – 1976), pp. 247-264). To say that Craig affirms simple refutation doesn't seem to be reflected in the footnote. I stand corrected if I missed another part of the book. But it seems to me that all Craig is saying is that Hart argues that the potential infinite 'need not imply' an actual infinite. That seems to be a lot different than claiming that Hart demonstrates the falsity that 'all' potential infinities imply actual infinities.
2. Lack of clarity - Again, if I've missed something in the book, much of what I'll say is off the mark. I agree that Hart thinks the idea of a potential infinite 'is not settled mathematical canon'. But again: Craig's citation of Hart is merely to support the point that potential infinities 'need not' imply actual infinities. Craig's reasons will obviously disagree with Hart's reasons for thinking that the idea's 'not being settled mathematical canon' is a good enough reason for Craig to discount the way he's motivated the idea of a potential infinite.
3. The temporal model - I fail to see how Craig's idea that the series of events is 'successively ordered' (happening one after the other in an 'isochronous sense') means the same as Hart's idea that series of numbers is 'isomorphic' to the natural numbers. For the successively ordered events to be isomorphic to the natural numbers presupposes that the future is actually infinite. It seems to me that it's possible for the future to be potentially infinite (in a way that's not actually infinite) and for such events to be successively ordered in an isochronous sense. But it doesn't seem to be possible for the series of moments to be isomorphic to the natural numbers and, at the same time, affirm that the series of moments is potentially infinite (in a way that's not actually infinite). Hence, Alex's point that Craig's idea fits Hart's idea 'pretty closely' isn't close enough for me.
I agree that 'at no point will you have written down an actual infinity of digits'. But then Alex cites Hart saying there's 'no theory of processes' in which to embed the idea of a potential infinite. Hence, what's sauce for the goose is sauce for the gander. Does Alex agree with Hart that the potential infinite is not settled mathematical canon? Does Alex agree with Hart that there's no theory of processes in which to embed the idea of a potential infinite? If that's the case, then how can Alex argue that we know enough about the potential infinite to know that Cantor's Intuition is the case? It doesn't seem that Alex could consistently argue that Cantor's Intuition applies to time if Alex agrees with Hart about the conceptual nebulousness of the idea of a potential infinite.
Alex does have Hart saying that it's not clear whether Cantor's Thesis applies to time, but Craig's footnote (citing Hart) doesn't say anything to the effect that Hart doesn't say this. I just don't see Hart is being this indispensable pillar the removal of which brings Craig's entire justification (regarding his belief in the potential infinite that doesn't imply an actual infinite regarding 'time') tumbling down.
4. Conclusion - Much of this blog is defeated if I've missed the citations of Hart Alex alleges are in Craig's book. Again: if I've missed this, I retract everything.
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