I have always appreciated the power of mathematics but,
unfortunately, am not very good at it. Of
course, in life all you have to know to succeed is a few more pages in the math
book than the next guy.
One thing that impressed me was the notion that there was
more than one "infinity". Like
most people, I understood infinity to be connected with the number of positive
integers. But on various occasions, things seemed not so clear.
One problem is how to determine when two sets of things have
the same number of elements (those of you who know math will have to excuse my
loose language). The example I use is to
think of two sets of things in two separate bags. Suppose I draw one item from one bag and match
it to one item from the other bag and I never run out of things in the first
bag before I run out of things in the second bag. Well, I think that is a good way to define the
number of things in the second bag as no greater than the number of things in
the first bag. If, for each thing in the
first bag, there is a unique assignment of a thing in the second bag, then I
think that is a good way to define the number of things in the two bags as the
same.
So, is the number of positive integers less than the number
of positive and negative integers? The
above rule for determining the answer shows that the number of positive integers
is the same as the number of positive and negative integers. All you have to do is start counting the
positive and negative integers using only the positive integers. The correspondence goes like this 1 goes with
1, 2 goes with -1, 3 goes with 2, 4 goes with -2, and so on. It is clear that the positive integers will
never run out and every positive and negative integer will have a unique
positive integer assigned to it. Yes,
Virginia, twice the number of positive integers is the same as once the number
of positive integers.
Oy.
It gets worse. Consider the points in the plane
characterized by two integers, such as (3,5). Does the number of these points in the
infinite plane exceed the number of positive integers? Not if we can count them systematically with the
positive integers. And we can. Start at (0,1) on the x axis and travel in a
sort of spiral about the origin. First, go
to (1,1), then (0,1), then to (-1,1), then to (-1,0), then to (-1,-1), then to
(0,-1), then to (1,-1). We have used the positive integers from 1 to 8 to go in
a "circle" around the origin. Next move outward to (0,2) and complete the
next largest circle. Clearly we will
never run out of the positive integers but will end up counting every pair of
positive and negative integers.
We would ordinarily think of the number of pairs of positive
and negative integers as infinity times infinity. But this exercise shows that our suspected
infinity times infinity is just infinity.
So, if we define ordinary infinity as the number of positive
integers, is that what we are stuck with? It seems not, because we can find a bag with
so many elements that they cannot be counted with the positive integers. The example I am thinking of is the set of infinite
decimals lying between 0 an 1 (inclusive).
The classic approach to showing that this set cannot be
counted with the positive integers is to first assume that it has been counted
and a two column table has been laid out with the positive integers in the
first column and the corresponding infinite decimals in the second column. Next, consider the following infinite decimal.
Its first digit differs from the first
digit of the first infinite decimal. Its
second digit differs from the second digit of the second infinite decimal, and
so on. Clearly this is a new infinite
decimal that nowhere appear in column two of our table. This contradicts the initial assumption that a
correspondence between the positive integers and the infinite decimals between
0 and 1 had been achieved.
We ran out of positive integers before we ran out of infinite
decimals between 0 and 1.
The number of infinite decimals exceeds the number of
positive integers.
Oy, again.
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