numbered events should have fewer events in it. But in fact it does not, as can be seen by
writing the two series one on top of the other:
1 2 3 4 15 16 17
2 4 6 8 10 12 14
and noting that each member of the top series corresponds precisely to each member of
the bottom series. Hence, the series of even-numbered events is both smaller and not
smaller than the upper series. These arguments against an actual infinity, however, are all
based on a confusion between two notions of “bigger than.” One notion is numerical: a
set is bigger than another if it has a greater number of members. The other notion is in
terms of part-to-whole relations: a whole is bigger than any proper part. When dealing
with finite quantities, anything that is bigger in the part-to-whole sense is also bigger in
the numerical sense. But this is not so in the case of infinite quantities. Although in the
part-to-whole sense there are more people in the hotel after a new guest arrives and
end p.121
there are more members of the original series of events, in the numerical sense there are
not. Indeed, mathematicians take the failure of the part-to-whole sense of “bigger than” to
imply the numerical sense to be the defining feature of infinity.
Alternately, the Kalam arguer may make use of modern scientific theories, such as that of
the Big Bang. However, in those cases, the argument is still subject to the possibility that
the theories will turn out to be false, or that it will turn out that there is a prior physical
cause of some sort to the Big Bang.
Probably the most powerful of the traditional cosmological arguments, as it involves the
least amount of conceptual baggage and controversial assumptions, is the one given by
Newton's follower Samuel Clarke (1705) at the beginning of the eighteenth century. Like
the Kalam argument, it begins with the contingent existential fact that there now exists an
aggregate of all the contingent beings there are, but unlike this argument, it does not have
to invoke any controversial claims about the impossibility of infinite aggregates. It
demands an explanation for the existence of this universe on the basis of a more general
version of the PSR than the one employed in the Kalam argument, namely, that there is
an explanation for the existence of every contingent being, even if it always existed. For
explanatory purposes, the universe itself counts as a contingent being, since it is an
aggregate of all the contingent beings there are. It therefore must have a causal explainer.
This cause cannot be a contingent being. For if a contingent were to be the cause, it
would have to be a cause of every one of the aggregate's constituents. But since every
contingent being is included in this aggregate, it would have to be a cause of itself, which
is impossible. The cause, therefore, must be some individual outside the aggregate; and,
since an impossible individual cannot cause anything, it must be a necessary being that
serves as the causal explainer of the aggregate. This holds whether the aggregate contains
a finite or an infinite number of contingent beings. Even if there were to be, as is possible
for Clarke, an infinite past succession of contingent beings, each causing the existence of