Infinite Series
1.
Concept and distinctions
In general, an infinite series is a
series without an ending term, like the series of natural numbers, 1, 2, 3…
Mathematician and logician Georg Cantor, defined it more precisely as a series
that has the same number of terms as one of its subseries. For example, the
series 1, 2, 3…, the series of natural numbers, has a subseries 2, 4, 6…, the
series of even numbers. However, there are just as many even numbers as there
are natural numbers, paradoxical as that may sound. That paradox identifies the
series as infinite.
Among infinite series, we may
distinguish between actual and potential infinites. The set of natural
numbers is an actual infinite: that set actually contains an infinite number of
members. A potential infinite, however, is a series that approaches an infinite
number but never reaches that point, as when we try to list all the natural
numbers, one by one, or when we divide an object by half, and then by half
again, and so on. In those cases we never reach an ending point, a last member
of the series. We never reach a number that we could call “infinity.”
2.
Apologetic Importance
Some forms of the cosmological
argument for the existence of God deny the existence of certain kinds of
infinite series. Thomas Aquinas, in the first three of his “five ways,” denied
that chains of causes (causes of motion, being, and necessity, respectively)
can go back forever. He argued that every causal chain has a beginning: a first
mover, a first cause of being, and a first necessary being, namely God. (See
Aquinas, Summa Theologica, Part 1,
Question 2, Article 3.) The Kalam
argument of Al-Ghazali, recently expounded by William Lane Craig, denies that
there can be an actually infinite series of events succeeding one another in
time. Therefore, the universe had a beginning, which must be explained by a
divine cause.
Craig argues, first, that there
cannot be an actually infinite collection of things (though there can be actually infinite sets of numbers),
and, second, that even if such a collection were possible it could not be
achieved by adding one member after another, as must happen in a temporal
succession of events.
To show that there cannot be an
actually infinite collection of things, he refers to the paradoxes noted by
Cantor: (1) In an infinite series, the whole is equivalent to some of its
parts. (2) One can add members to an infinite set without increasing the number
of members in the set. (The number remains at infinity.) (3) One can remove
members from the set without decreasing its membership. Such is the case in the
abstract world of numbers. But, Craig says, it would be impossible to have a
set of concrete objects or a series of events that had these properties. He
uses the illustration of “Hilbert’s Hotel” from George Gamow, One, Two, Three, Infinity, p. 17: If a
hotel had an infinite number of rooms filled with guests, additional guests
could check in without anyone moving out, and the number of guests would be the
same as before. The sign could read, “NO VACANCY—GUESTS WELCOME” (Craig, Reasonable Faith, p. 96).
Then Craig argues that even if we
grant the possibility of an actually infinite collection of things, we cannot
form such a collection by adding one member after another. It is impossible,
for example, to count an infinite collection one by one; for “No matter how
many numbers you count, you can always add one more before arriving at
infinity” (p. 98). The same must be said of an infinite series of events in
time. If the process of nature and history extends infinitely far into the
past, then it is an infinite succession of events, and that succession has
proceeded one by one, ending precisely at the present moment. But why did it
end now, rather than yesterday, or a thousand years ago? For on this hypothesis
yesterday was also the end of an infinite chain of events, and so was the
moment a thousand years before the present one. But, in fact, there can be no
end at all, for an infinite series never ends. So, Craig concludes, the series
of past events is finite. Therefore the universe had a beginning, and therefore
a cause, because “whatever begins to exist has a cause” (p. 92).
3.
Evaluation
Certainly it is difficult to
conceive of an actually infinite collection of things. Hilbert’s Hotel is counter-intuitive;
but many find the Cantor paradoxes themselves hard to believe at first hearing.
After we learn to work with infinite sets of numbers, we tend to accept the
Cantor definitions as a matter of course. We have not, however, encountered
infinite sets of material objects. But if we ever do, might we not eventually
get used to the strange properties of such sets? Here, images are important.
The idea of an infinite hotel is somewhat ridiculous, as is, say, the idea of a
hotel with the hiccups. But how about the idea of an infinitely extended chain
of beads? Might we not one day get used to the idea of adding or subtracting
beads without changing the number in the infinite collection?
Part of the problem is that when we
try to picture in our minds an infinite hotel, we tend to think of it as a
finite hotel with very odd properties: people being squeezed into it without
others being squeezed out. But if the hotel were truly infinite, those
properties would not be odd, but expected, hard as it may be to imagine these
properties in a mental picture. It is also hard to imagine such properties in a
series of numbers; but Cantor proved that they exist.
Similarly, the notion of an
infinite series of events continuing through time is hard to comprehend, but is
it impossible? I agree with Craig that it is impossible to count through an
infinite series and end with a final number. But (1) if time itself were
subjective, rather than objective, then an infinite set of past events might
exist simultaneously (like the series 1, 2, 3...), rather than existing by a
temporal process of addition. (2) The same would be the case if time were an
objective dimension of n-dimensional space, and all events of past, present,
and future, could be viewed together by a being of a higher dimension. And (3)
if we could go backward in time from the present, then we could visit
yesterday, the day before yesterday, and the day before that, much as we now
move from today, to tomorrow, to the day after. In that case, we would perceive
the days of past history much as we now perceive the days of the future: as a
potential infinity, rather than an actual infinity.
Of course, these three suppositions
are contrary to Craig’s own theory of time. See his Time and Eternity: Exploring God’s Relationship to Time (
Thomas Aquinas would object to
supposition (3) that even a potentially infinite series of natural events in
the past is insufficient to account for the world as we know it. For on this
supposition, each event is caused by a previous one; no event actually begins
the series. Therefore, no event (or group of events, by the same logic) serves
as the cause of the rest. So the universe is uncaused, unexplained. Aquinas
believes the universe must have a cause; so the chain of causal explanation
cannot be infinite, even potentially infinite.
Aquinas argues that the universe
has a cause; therefore there cannot be an infinite series of causes. Craig
argues the reverse: there cannot be an infinite series of causes; therefore the
universe must have one cause. I confess that I find Aquinas more persuasive: it
seems more obvious to me that the universe requires a cause than that an
infinite series of events is impossible. But even Aquinas’ s view requires
assumptions, namely that nothing exists or happens without a sufficient cause,
and that causes (including the cause of the universe) are accessible to human
reason. Many skeptics of the past and present would not grant those
assumptions.
My conclusion is that our concepts
of cause, reason, and infinite series depend on worldviews, on ontological and
epistemological assumptions. They are insufficient in themselves to serve as
grounds for worldviews. A Christian theist will think differently from a
skeptic about these matters. His Christian theism will govern his concepts of
cause, reason, and infinity, rather than the reverse.
Bibliography
G. Cantor, Contributions to the Foundations of the Theory of Transfinite Numbers (ET, Chicago: 1915)
W. L. Craig, The Kalam Cosmological Argument (New York, 1979)
___, Reasonable Faith (Wheaton, IL, 1994)
G. Gamow, One, Two Three, Infinity (