The 'Quirk' Problem

In my "keeping it real" post below, I raised some objections to a standard presentation of the sort of kalam cosmological argument at someone else's site. The argument has as a crucial premise that an actual infinite in time cannot exist. The premise is supported by refering to some standard paradoxes of infinity. But since we have a well worked out system of transfinite mathemetics, it seemed ridiculous now to say that we cannot conceive of such an actual infinite consistently.

However, at Peter Suber's educational site, he makes reference to a problem for transfinite math which he calls the 'quirk' problem. His presentation is so clear I won't add anything to it. Just see the link. This problem is a problem which suggests that transfinite math is inconsistent. So it seems that defenders of the kalam argument are not on bad ground by resisting objections based on the conceivability of an actual infinite.

If he's right, at least I'll be able to rebut demands to eat an aleph-null amount of crow.