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:

a₁ = 0.a₁₁ a₁₂ a₁₃...
a₂ = 0.a₂₁ a₂₂ a₂₃...
a₃ = 0.a₃₁ a₃₂ a₃₃...

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

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.b₁ b₂ b₃..., 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 a₁ in the first decimal, from a₂ in the second decimal, from a₃ 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, Naïve 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.