On the Varieties of Finitism

Defenders of the Kalām Cosmological Argument appeal to the so-called
Hilbert’s Hotel Argument to establish the "nitude of the past based on the
impossibility of actual in"nites. Some of their opponents argue that this proves
too much because if the universe cannot be beginningless due to the impossibility of actual in"nites, then, for the same reason, it cannot be endless either.
Discussing four different senses of the existence of an actual in"nite, I criticize
both sides of the debate by showing, on the one hand, that the Hilbert’s Hotel
Argument is not powerful enough to rule out the possibility of the in"nitude
of the past and, on the other hand, that the soundness of the argument for the
"nitude of the past from the impossibility of actual in"nites does not establish
the soundness of the parallel argument for the "nitude of the future