William Lane Craig's The Kalam Cosmological Argument

1.  Whatever begins to exist has a cause.

2.  The universe began to exist.

First Arg for premise 2:

A.  An actually infinite number of things cannot exist.

An Actually Infinite Seriesis an existing set of things that is infinite in number.

A Potentially Infinite Seriesis a collection that is increasing towards infinity but never actually gets there.  It is a series that can have new members added to it indefinitely, but is not itself actually infinite.

i.  If an actually infinite number of things could exist, then it would "spawn all sorts of absurdities."

Hilbert's Hotel:

a.   Hypothesis:  Consider an actually infinite set of rooms, all occupied, in a hotel.

b.  It is possible to move the guests to even numbered rooms making an infinite number of rooms empty, and possible to add an infinite number of new guests.

b.  Infinite = number of guests before the new arrivals

c.  Infinite = number of guests after the new arrivals.

d.  The number of guests before the new arrivals = the number of guests after the new arrivals.

e.  But the number of guests before the new arrivals is not = the number of guests after the new arrivals.

f.   So the hypothesis that an actually infinite series is possible generates a contradiction.

ii.  Any hypothesis that generates absurdities is itself impossible.

iii.  Therefore it is impossible that an actually infinite number of things can exist.

B.  A beginningless series of events in time entails an actually infinite number of things.

C.  Therefore, a beginningless series of events in time cannot exist.

Second Arg. for Premise 2

A.  The series of events in time is a collection formed by adding one member after another.

B.  A collection formed by adding one member after another cannot be actually infinite.

C.  Therefore, the series of events in time cannot be actually infinite.

Conclusion from the two arguments for Premise 2:  if the series of events in time cannot actually be infinite, then they must have a finite starting point or a beginning.  So 2 is proven,The universe began to exist.

3.  Therefore, the universe has a cause.

4.  Philosophical analysis reveals that such a cause must have several of the principal theistic attributes.

5.  Therefore, the cause of the universe is God.

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