. Since we
are used to dealing concretely only with finite sets, such properties of infinite
sets do seem almost unbelievable. Nevertheless, as demonstrated
by Georg Cantor (1845-1918) and confirmed by modern mathematicians,
trans-finite mathematics is logically consistent. Although a number of
paradoxes can arise in the usage of infinite sets, these are generally a
result of self-reference problems.
Craig notes that he is arguing only against an actual infinity in the real
world, and in no way wishes to undermine the concept of actual infinity
as found in Cantorian transfinite mathematics. But there, according to
Craig, actual infinity is only an idea:
What I shall argue is that while the actual infinite may be a fruitful
and consistent concept in the mathematical realm, it cannot be translated
from the mathematical world into the real world, for this would
involve counter-intuitive absurdities.9
He gives a number of illustrations, one being a library consisting of an infinite
number of books. Such a library has some strange properties. For
example, if we eliminate half of the books (i.e., the odd-numbered ones)
then we still have as many books as we stayed with. Yet if we now loan
out the remaining books (i.e., the even-numbered ones) then we have
nothing left, even though we have removed exactly the same number as
before. Furthermore, Craig argues, if we add a book to an infinite library
then we can see that the collection is increased by one; we don't have the
same number of books as before, as should be the case for an infinite set.
He concludes that such examples serve to illustrate that an actual infinite
cannot exist in the real world.
Does this argument really demonstrate the impossibility of an actual
infinity in the real world? I think not. The above operations on books
could also be done on, say, the positive integers. Remove the odd ones
and you still have an infinity left; remove now also the even ones and
nothing remains. Or take the even integers, which form an infinite set,
and you can still add any odd number to it, leaving us still with an infinite
set. If these operations are permissible for numbers, why not also for
books? Since one could set up a one-to-one correspondence between integers
and books in a library (or events in time), it follows that likewise
an actual infinity of books (or events) need not involve any logical difficulties,
however counter-intuitive such an actual infinite library might
be.
The essential distinction between books and numbers, according to
Craig, is that the former are concrete things while the latter are mere
ideas. We can see and feel books and hence, presumably, counter-
intuitive notions must be rejected. Yet one might question whether it is
meaningful for us to even contemplate the occurrence of an actual infinity
in the real world of our experiences. After all, within the limitations of
our finite experiences, memories, and thoughts we can never distinguish
between the infinite and merely the very large. Even if we could
eliminate half of the books in an infinite library, the remainder will sell
be beyond our ability to count and thus, in any practical sense, will be the
same as what we started with. Since all our human experiences are finite,
the real actual infinite is necessarily counter-intuitive. But that by itself
does not render it impossible.
God and the Future
Suppose, for the sake of the argument, that the proofs against the actual
infinite were valid. Such a ban on the actual infinite would have
some awkward theological consequences. Consider, first of all, the question
of future events. The Bible, in its description of the life-hereafter,
pictures it as a temporal existence, with flowing water and ripening fruit,
where the saints shall reign for ever and ever (Rev.22:1-5): a seemingly
endless future time. Is this view of the future not ruled out by the arguments
against an actual infinity of past events, which imply that the future
must likewise be finite?
Craig argues that this need not be the case. The future differs from the
past in that it is not an actual completed infinity; it is only potentially infinite,
in the sense of being inexhaustible. A potential infinity is, according
to Craig, permissible:
a potential infinite is a collection that is increasing without limit but is
at all times finite…it is not truly infinite - it is simply indefinite.10
Past events are real since they have actually occurred; future events,
asserts Craig, do not actually exist because they have not (yet) occurred.
At fast sight such a past/future distinction seems valid enough.
However, when applied to the omniscient God of orthodox theology (and
it must be kept in mind that this is just a step in an alleged proof for the
existence of such a God), it becomes problematic. After all, such a God
knows the future as definitely as He knows the past. If the future is
indeed endless, then to an omniscient God it exists as a definite actual infinity,
rather than as an indefinite potential infinity. It would seem that to
God an endless future would have the same status as a beginningless past.
Both entail the notion that God has an infinite stock of memories or
thoughts (cf. Ps. 139:17).
Thus the considerations leading to a finite past must similarly apply to
the future. If Crag's argument against an actual infinity is valid it implies
that God's knowledge encompasses only a finite number of future events.
This leads to the conclusion that either the future is finite, and there is a
last event, or God's knowledge of the future is incomplete.
God and the Past
There are further difficulties. Crag's arguments against the real existence
of actual infinity are sufficiency general that they seem to apply not
only to the physical universe, but also to God Himself. This must have
significant implications for God, who is generally considered in orthodox
theology to be infinite.
For example, Crag's ban on the actual infinite implies that also God's
past must then be finite. Indeed, Craig himself comes to this conclusion:
… prior to creation God would have to be changeless. Otherwise you
would get an infinite series of past events in God's life, and we have
seen that such an infinite series is impossible. So God would be
changeless and hence, timeless prior to creation.11
It seems that Craig views God as being essentially “frozen” for an infinite
time before the fast event. This seems to place rather stringent conditions
on the nature of God, ruling out any succession of acts or thoughts prior
to creation.
It is interesting that Craig applies the prohibition of an actual infinity
only to events, and not to the passage of time itself. He does allow for a
past infinity of time. Is it permissible to allow an infinity of past time
while ruling out an infinity of past events (i.e., an eternal “frozen”
universe)? Even then there would still be the everlasting passage from
one time unit to the next, else how is infinite time to be distinguished
from an isolated unit of time?
It could be argued that, since there are no physical events to measure
this time, we are not confronted with an actual infinity. But does this
limitation apply to God? If God is omniscient then He must know also of
the passage of time. To God, it would seem, even the passage of a unit of
time is something “which happens” and thus should count as an event. If
God has existed through a past infinity of time then an actual infinity of
units of time has elapsed. It follows that God's knowledge of a past that
consists of finite events embedded within infinite time must include
knowledge of an actual infinity of past units of time.
Furthermore, the admission that time itself has a finite past endangers
Craig's argument for a personal creator. Craig's argument for the first
cause being a personal agent rests upon the assumption that there was an
infinite timespan before creation. Since all moments in such an infinite
time are alike, he argues, it requires a personal being to freely choose to
create at any time wholly apart from any distinguishing conditions of one
moment from another.12 Elsewhere Craig affirms that the only way to
have an eternal cause but an effect that begins at a point in time is if the
cause is a personal agent who freely decides to create an effect in time.
For example, a man sitting from eternity may will to get up.13
The conclusion to which we are led, assuming that the actual infinite
can't exist, is that the universe, time, and God all started to exist a finite
time past.
To summarize, a ban on the actual infinite seems to place undue
restrictions not only on the past, but also on the future, on God, and on
time itself. Although there may be aspects of infinity that appear incomprehensible
to us, it seems to me that, in the absence of water-tight logical
disproofs of actual infinity, the better course is to attribute this per
ceived deficiency to human finiteness rather than to confer undue constraints
upon God and His attributes.
For the cosmological argument to work without placing unwarranted
limitations on God it is necessary to make a proper distinction between
Creator and creation. The demonstration of the finite past of the physical
universe is therefore perhaps better based on physical, rather than logical,
grounds. Two main lines of evidence are often cited as scientific proof
for the beginning of the universe: the big bang singularity and the second
law of thermodynamics.
It is beyond the scope of this paper to examine in detail the alleged
scientific evidence for a beginning of the universe. Since such scientific
conclusions must go beyond current observations, they are necessarily
speculative: it is possible to devise many plausible cosmological models
that avoid a beginning in time. Even if this were not the case, if only a
finite-past Big Bang cosmology were scientifically viable, this would still
involve unverifiable assumptions such as, for example, that of the universal
applicability (always and everywhere) of the physical laws observed
here and now. How, for example, can we be sure that no miracles have
occurred? Certainly, as a step in a proof for the existence of the Biblical
God, any scientific proof of a beginning of the universe is unsatisfactory
since an omniscient and omnipotent God could well have interacted with
the universe, perhaps in miraculous ways, so as to avoid the limitations of
thermodynamics and space-time singularities. The best one could do is
argue that any theistic implications of naturalistic big bang cosmology
imply, not the truth of big bang cosmology but rather the deficiency of
naturalism.
In short, the case for the existence of God based upon purely logical
or scientific proofs for a finite past of the universe are ultimately self-
defeating: a ban on the actual infinite places undue constraints on God;
scientific proof for a beginning assumes the absence of supernatural
events.
NOTES AND REFERENCES
1
An earlier version of this material is being published as part of a
longer paper “On Craig's Defence of the Kalam Cosmological Argument"
in Proceedings of the Pascal Centre Internation Conference on
Science and Belief, (held August 11-15, 1992, Redeemer College.
2
Norman Geisler & Winfred Corduan, Philosophy of Religion (Grand
Rapids: Baker Book House, 1988, 2nd ed.), p.l50.
3
See Richard Sorabji, Eme, Creation and the Continuum (Ithaca: Cornell
University Press, 1983), p.l98.
4
See Herbert A. Davidson, Proofs for Eternity, Creation and the Existence
of God in Medieval Islanuc and Jewish Philosophy (Oxford:
Oxford University Press, 1987), pp. 117-153.
5
William L. Craig, The Kalam Cosmological Argument (London:
Macmillan, 1979).
6
J. P. Moreland, The Creation Hypothesis (Downers Grove: lnterVarsity
Press, 1994), pp. 18-23.
7
See, for example, J. P. Moreland, op. cit., p. l9.
8
Quentin Smith, 1987. “Infinity and the Past”, Philosophy of Science
54:63-75.
9
Craig, op. cit., p. 69.
10
William L. Craig, 1980. “Philosophical and Scientific Pointers to
Creation Ex Nihilo”, Journal of the American Scientific Affiliation,
32:6.
11
Craig, op. cit. p. l2.
12
Craig (1979), p. 151.
13
William L. Craig, 1984. Apologetics: An Introduction (Chicago:
Moody Press), p. 93.