Here’s the problem, it seems to me: in order for the collection to be completed, we must have already enumerated, one at a time, an infinite number of previous cards. But before the final card could be added, the card immediately prior to it would have to be added; and before that card could be added, the card immediately prior to it would have to be added; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any card to be added to the collection.
This way of putting the argument is somewhat akin to Zeno’s argument that before Achilles could cross the stadium, he would have to cross half-way; but before he could cross half-way, he would have to cross a quarter of the way; but before he could cross a quarter of the way, he would have to cross an eighth of the way, and so on to infinity. Therefore, Achilles could not arrive at any point. Zeno’s paradox is resolved by noting that the intervals traversed by Achilles are potential and unequal. Zeno gratuitously assumes that any finite interval is composed of an infinite number of points, whereas Zeno’s opponents, like Aristotle, take the interval as a whole to be conceptually prior to any divisions which we might make in it. Moreover, Zeno’s intervals, being unequal, add up to a merely finite distance. By contrast, in the case of an infinite past the intervals are actual and equal and add up to an infinite distance.
About the best that the critic of the argument can do at this point, I think, is to say that if one adds cards at a rate of, say, one card per second, then the collection can be completed because there has been an infinite number of seconds in the beginningless past. But clearly this response only pushes the problem back a notch: for the question then is, how can the infinite collection of past seconds be formed by successive addition? For before the present second could elapse, the one before it would have to elapse, and so on, as before. Because the problem is applicable to time itself, it cannot be resolved by appealing to infinite past time.