Infinity:
What has no end, no boundary, no limit. The infinite or infinity has
been the subject of much speculation in philosophy and
mathematics, and also
in theology, where it has been widely considered to be one of
God's
attributes.
The definition I proposed is like the Greek, who used the term 'to apeiron'
for it  the boundless  and speculated quite a lot about it.
It is quite easy to imagine that one can take one more step, or divide
something one more time, and never stop doing this. Then what? Where does it
end? Can it end? Can it fail to end? How does one know? Aristotle has an
interesting and subtle discussion, but no really precise analysis, and no
mathematics of infinity.
By far the most precise analysis of the concept of infinity, certainly
mathematically speaking, was first made by Georg Cantor, who also developed
set theory to provide an analysis of it.
A settheoretical definition of an infinite set is: A set which has a proper
subset that is as large as the set itself. A proper subset of a set is a
subset of a set that lacks at least one element
that is in the set. The reason for the definition may be presented as
follows (using a sort of argument Galileo also knew):
1
2 3 4 5 ... n ... and so on
2 4
6 8 10 .. 2n ... and so on
The first line is the beginning of a full listing of the natural numbers;
the second line is the beginning of a full listing of the even natural
numbers. Now it is clear that for any number in the first list, there
corresponds precisely one double of it in the second list, and to every number
in the second list, there corresponds precisely one half of it in the first
list, and indeed one can combine both into one list of pairs thus:
(1,2) (2,4) (3,6)
... (n, 2n) ... and so on
Now, if one supposes that two sets have the same number iff there is
a 11
correspondence between them, then the argument just given strongly suggests
there are precisely as many natural numbers as there are even natural numbers
 supposing that either collection indeed has a totality, that can be
meaningfully said to have a number, which is what Cantor supposed.
This shows, accordingly, on those Cantorian assumptions, also a way in
which infinite sets  as defined  differ from finite sets, for the number of
a proper subset of a finite set is always smaller than the number of the set
it is a proper subset of.
And to conclude the argument: Clearly, the even natural numbers are a
proper subset of the natural numbers, in the sense that each even natural
number is a natural number, whereas some natural numbers  all the odd ones,
indeed  are not among the even natural numbers.
Consequently, at least plausibly, the natural numbers are an infinite set,
for it contains a proper subset that has precisely as many numbers as it, and
so is as large as it. (The reader should note that the weakest point of the
above argument is the assumption that an infinite set has a totality, that has a number.)
All of this was meant to illustrate the sort of reasoning that led Cantor
to his settheoretical definition of infinity, but it should be remarked that in the case of
the natural numbers one can do without: These are also infinite in sense that
there is no largest (finite) natural number, since one always can get another
one that is still larger by adding 1 to the hitherto largest. And this is the
sense of limitlessness the Greeks already knew and pondered. Here it is not
assumed that what is limitless has a totality and a number.
Cantor was a mathematician, and part of his reasons (though not all, since
he also had philosophical and theological concerns) was to provide a clear and
cogent foundation for the many arguments in mathematics that somehow involve
the notion of the infinite, namely in the calculus (differential and
integral); in geometry (projective, especially); with series, like 1/2, 1/4,
... , 1/2^{n} and their sums and convergences (the series just given intuitively
sums to 1 and converges towards 0, all "in the limit", "after an infinity of
terms"  (*)); and indeed with the natural and real numbers, of which there seem to
be infinitely many, and about which Cantor had an ingenious argument to prove
that even so there are more real numbers than there are natural
numbers.
A good introduction to the many problems involved in the notion is A.W.
Moore's "The Infinite", that concerns both the mathematical and the
philosophical notions of the infinite, has mostly clear explanations, and
defends a kind of finitism.
Indeed, one good argument to show that Cantorian infinities are a bit
difficult to swallow intuitively is that by the axiom of the power set of
standard set theory there always is a
power set of any given
set, which by an argument of Cantor must always be greater than the set
itself. Hence there are infinities upon infinities  which, for one thing, is
more than is needed for classical
mathematics, for that can do with such infinities as are involved in the
natural and real numbers, that are  or seem to be  the smallest infinities
in the Cantorian hierarchy of infinities, implied by the power set axiom.
A good introduction to how infinity is dealt with in standard mathematics
for the calculus and the theory of functions, mostly without set theory, is
Konrad Knopp's "Infinite sequences and series". Any good
introduction to set theory shows how
infinities are reckoned with there. Any calculus text presumes some theory of
infinity, though usually this is not clearly and fully articulated in such
texts (in which this may not be necessary either).
(*) To show the sort of ordinary mathematical reasoning
with or around infinities, here are proofs of the statements that the infinite
series 1/2, 1/4, ... , 1/2^{n}, .... sums to 1 and converges
towards 0:
A. Suppose 1/2 + 1/4 + ... + 1/2^{n }+ .... ad infinitum has
a sum. Call it S. Then we have 1/2 + 1/4 + ... + 1/2^{n }+ ....
= S. Multiply all terms of this equation by 2 (and suppose you can do this an
infinity of times). The result must be 1 + 1/2 + 1/4 + ... + 1/2^{n }
+ .... = 2S which is to say 1+S=2S which is to say S=1. Qed.
B. Let me have all of 1/2, 1/4, ... ,1/2^{n}, .... ad infinitum.
Now if anyone chooses any small nonzero difference or distance, as small as
he pleases, from 0, then I can produce a number from the series which is even
smaller. So I can approximate 0 as closely as I wish, or he wishes, and within
any arbitrarily assigned small distance. Qed.
One other reason for this footnote is to show the sort of arguments
ordinary mathematicians used. Both arguments can be found in Leibniz, in the
17th C, when founding the calculus, if not precisely in these terms.
It was this sort of reasoning Cantor wanted to make really precise, and for
this reason he found or created the foundations of
set theory and indeed introduced the
concept set.
