Some
remarks on infinity

There are two basic characterizations of infinity: *potential*
infinity and *actual* infinity. Take a set with an infinity of
objects in it, e.g., the set N of natural numbers {1, 2, 3,....}. One can
characterize N as a potentially infinite set, that is, a set which is open in
the sense of being incomplete: given any number of elements in it, it's always
possible to *add another one* (we are not going to run out of natural
numbers). However, one can think of N as an actually infinite set, that
is, as a set which is *actually completed*, with all the natural numbers
in it.

Historically, mathematicians and philosophers have preferred
to restrict their considerations to potential infinity because some of the
implications of actual infinity are at odds with seemingly obvious
truths. To see why this is so, one has to understand the notion of
biunivocal (or one-to-one) correspondence.

There is a biunivocal correspondence between two sets A and
B if and only if for *each and every* member of A there is *one and only
one* member of B. For example, consider two sets, A = {1, 2, 3) and B
= {2, 4, 6}. It's possible to establish a biunivocal correspondence
between them (just associate each member of A with its double in B).
Then, we can say that there are as many members in A as in B (if there is a
chair for each guest and a guest for each chair, then there are as many chairs
as there are guests). Consider now N, the actually infinite set of
natural numbers {1, 2, 3, 4...}, and E, the actually infinite set even numbers
{2, 4, 6, 8...}. Then there is a biunivocal correspondence between N and
E. Hence, N and E have the same number of members, that is, the same __cardinality__.
This result in incompatible with the principle P that the whole (in this case,
N) is always larger than the part (in this case, E). N has all the
members of E, plus all the odd numbers; so, how can it possibly have as many
members as E, its part? Aristotle, just to mention one philosopher, was
unwilling to give up P, and consequently rejected the idea of actual infinity
as incoherent. Mathematicians agreed, essentially banning actual infinity
from mathematical demonstration (consider, for example, the development of the
method of exhaustion by Eudoxus). This state of affairs changed in 1872,
when Dedekind introduced the notion of an infinite set as one which can be put
in biunivocal correspondence with one of its proper subsets (a proper subset is
one which is not identical to the original set. More on this in a
moment).

Dealing with infinite sets requires, among other things,
distinguishing between two sense of 'greater than' or 'more than'.
Consider the set A={1,2,3,4} and the sets B={1,2,3}. Then, since all the
members of B are also members of A but not all members of A are members of B, B
is said to be properly __included__ in A or, which is the same, to be a
proper subset of A. Notice that since both A and B are finite (each
contains only a finite number of members), B has fewer members than A in the
sense that any attempt to establish a biunivocal correspondence between A and B
must fail because there will always be some members of A to which no member of
B corresponds. However, as we saw, there is a biunivocal correspondence
between N and E even if E is a proper subset of N. So, we should say that
there are 'more' members in N than in E in the __inclusion__ sense but not
in the __cardinality__ sense.

A natural question is whether *all* infinite sets have
the same number of elements. Clearly some do; for example, E, the set of
odd numbers, the set of prime numbers, the set of natural numbers greater than
235, can all be put in biunivocal correspondence with N. Hence, they have
the same number of elements as N, that is, they are *countable* sets.

N seems to be much smaller than the set R of all rational
fractions (fractions whose numerator and denominator are natural
numbers). The reason is that R has the property of *density*, that
is, between any two rational numbers there is a third one. [Proof: let a and b
be two rational numbers; then (a+b)/2 is between a and b and belongs to
R). So, between any two rational fractions there is an infinity of
other rational fractions; by contrast, between anay two succesiive natural numbers,
e.g., 5 and 6, there is no other natural number. Consequently,, it would look
as if it's impossible to establish a biunivocal correspondence between N and
R. However, in 1874 Cantor showed that it is not so: N and R have the
same cardinality or, as he put it, they have the same *power*. To
see this, consider the picture below. Arrange rational fractions so that
all the fractions with 1 as denominator are in the first line; all those with 2
as a denominator are in the second line, etc. Then, all the rational
fractions will be in the grid. Now, follow the arrows in the figure,
associating 1/1 with 1; 2/1 with 2; 1/2 with 3; 1/3 with 4, etc. Then all
the fractions can be counted, that is, there's a biunivocal correspondence
between R and N.

One begins to wonder if all sets of numbers have the same
power, but Cantor proved that this is not the case. The set R of all real
numbers (rational numbers plus irrational numbers, e.g. the square root of 2
or pi, the ratio between a circumference and its radius), for example, has a
higher power than the set of rational fractions or natural numbers. To
show this, Cantor used a *reductio ad absurdum*. Real numbers can be
expressed as non terminating decimals (so that 1/3, for example, appears as
0.333..., 1/2 as 0.499..., and so on). Suppose, then, that the set of
real numbers between 0 and 1 is countable. Then, they can be arranged in
a denumerable order:

a_{1} = 0.a_{11} a_{12} a_{13}...

a_{2 }= 0.a_{21} a_{22} a_{23}...

a_{3} = 0.a_{31} a_{32} a_{33}...

..........................

where a_{ij} is a digit between 0 and 9 inclusive. To show
that *not all* of the real numbers between 0 and 1 are included above,
Cantor produced an infinite decimal different from all of those listed.
He formed the decimal b = 0.b_{1} b_{2} b_{3}..., where
b_{k} = 9 if a_{kk} is 1 and b_{kk} = 1 if a_{kk}
is not 1. The real number b will be between 0 and 1 and yet it will be
unequal to any one of those in the arrangement supposed to contain all of the
real numbers between 0 and 1. For, it will be different from a_{1}
in the first decimal, from a_{2} in the second decimal, from a_{3}
in the third decimal etc. So, there are more real numbers between 0 and 1
than there are natural numbers. Hence, there are more numbers in R than
in N.

Cantor also proved that given any set A, P(A), the power set
of A, that is, the set which has as elements all the subsets of A, has larger
power than A. For example, let A be {1,2,3}; then the set B ={ {1}, {2},
{3}, {1,2}, {1,3}, {2,3}, {1,2,3}, the empty set } has more elements than A
(this is an example, not a proof!) Hence, P(R) has greater power than R;
P(P(R)) has greater power than P(A), etc. In short, there is a *hierarchy* of
infinities. For more on these issues, read a book on set theory like
Halmos, *Naοve Set Theory*

Distinction between open and closed intervals

Consider the following ordered sets of numbers:

A={all the rational numbers x such that 3<x<5}

B={all the rational numbers x such that 3 < =x<5} ('< =' means
"less than or equal to")

It's important to understand that A and B are not identical because A does
not contain 3 but B does. That is, B contains a smallest member,
namely 3, but A does not because given any number y such that 3<y<5, it
is always possible to find another number z such that 3<z<y (Simply let z
= (3+y)/2). A is called an open interval because it does not contain any
endpoint (neither 3 nor 5 are in A). This fact is represented as A = (3,5).
By contrast, B is a semi open interval because it contains only one endpoint,
namely 3, while the other endpoint (namely, 5) is not in B. This fact is
represented as B = [3,5). Notice that a parenthesis is used to designate
the endpoint which makes the interval open, and a bracket to designate the
endpoint which makes the interval closed. So, (3, 5) is open at both
ends, or more briefly, it is an open interval; [3,5) and (3,5) are both semi
open intervals; [3,5] is a closed interval.

Notice that an interval such as (3,5) has neither a first nor a last member,
and yet none of its members is smaller or equal to 3 or greater or equal to
5. So, even if time doesn't have a first moment, it doesn't follow that
the past is infinite as long as time is dense, i.e., is such that between any
two instants there is always a third. Analogously, even if time doesn't
have a last moment, it doesn't follow that the future is infinite as long as
time is dense.