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© Copyright 2003, Jim Loy
Disclaimer: This page deals with sets, which are just collections or lists. This set has three elements: {5, 17, 4}; this set has four elements: {John, Paul, George, Ringo}. This set has no elements, and is called the empty set {}. Don't be too concerned with the details of set theory here. I will write about it later. This article shouldn't be very difficult without detailed knowledge of set theory.
In my mathematics pages, I have repeatedly warned that infinity is not a number, and that treating infinity as a number leads to trouble (mostly wrong answers). Some of my readers have probably shaken their heads at my ignorance, because they know that Georg Cantor showed that infinity is a number, and that there are several infinities of varying sizes. Let me repeat that infinity is not a number, and that treating infinity as a number leads to trouble (mostly wrong answers). But, it is extremely productive to consider the idea of infinity as a number, and see where it leads. We still need to be careful to never treat infinity as a number in most branches of mathematics. In this one branch of mathematics, the study of transfinite numbers, we will treat infinity as a number. And we will get some interesting and valid results, which we can apply to other branches of mathematics.
The study of transfinite numbers is a branch of set theory. We begin by asking, "How do we count?" This is our invitation to get down to the real basics of numbers. The answer to that question is this: We count by establishing a one-to-one correspondence between a set and the set of natural (counting) numbers. Let's say that a certain set has three elements. How do we know that? Well we compare it with this set {1, 2, 3}, and we see that the first element can be matched up with 1, the second can be matched up with 2, and the third can be matched up with 3. It has 3 elements. But you retort, I just count the three elements; I don't establish any one-to-one correspondence. Yes you did. Counting is precisely that, matching up an element with 1, another element with 2, etc. A very interesting thing about this is that it works for infinite sets.
Let's count the elements in the set of even positive integers. We do that by comparing the even numbers with the counting numbers:
1 2 3 4 5 6 7 8 9 10 11 . . . 2 4 6 8 10 12 14 16 18 20 22 . . .
This correspondence goes on forever. And we can find no even number that does not match up with a counting number. We can only deduce that the two sets have the same number of elements. This may seem counter-intuitive, but it is straightforward. The set of even positive integers has the same number of elements as the set of counting numbers. They are both infinite sets of the same size. The set of all integers (positive and negative) matches up the same way. So does the set of all integers ending in 000. These are all countable sets, because they have the same number of elements as the counting numbers.
These countable sets are represented by the symbol
, or
aleph null. This symbol can be taken to be a number (for our purposes here), a
number greater than any real number, a transfinite number. It is the
number of elements in the set of positive integers.
is the
cardinality of the set of positive integers. We find that:
r
These last two equations may require some thought.
is the
number of elements in the set of pairs of counting numbers. An example of such
a set is the set of rational numbers. Is it possible that the set of rational
numbers is countable? Here is the set of positive rational numbers, with some
duplication (as I will not be reducing them to their lowest terms):
1/1 1/2 1/3 1/4 1/5 1/6 . . . 2/1 2/2 2/3 2/4 2/5 2/6 . . . 3/1 3/2 3/3 3/4 3/5 3/6 . . . 4/1 4/2 4/3 4/4 4/5 4/6 . . . . . .
We can count these merely by (1) starting in the upper left corner, (2) move right, (3) move down, (4) move left, (5) move down, (6) move right, (7) move right, (8) move up, (9) move up, (10) move right, etc. We get this list of fractions: 1/1, 1/2, 2/2, 2/1, 3/1, 3/2, 3/3, 2/3, 1/3, 1/4, etc. And so the rational numbers are countable, merely because we can count them.
How about the real numbers, which are made up of the rational numbers and the irrational numbers (like pi or the square root of 2)? Surely the reals are countable, too; how many irrational numbers can there be? It turns out that there are many irrationals indeed. The set of reals is not countable. Let's list them all as decimal fractions:
0.bcdefg... 0.ijklmn... 0.opqrst... . . .
Here are three numbers between 0 and 1. Let's pretend that we have listed all real numbers between 0 and 1, without duplications (0.70000... is listed, but 0.69999... is not, as they are the same number). Well, let's create a real number that is not on our list. We will start with "0." and then choose a digit other than "b" for the next digit: 0.u. This number is definitely not equal to our first real number in the list. The next digit will be v, different from j in our second real number. The next digit will be w, different from q in our third real number (0.uvw). We keep doing this for all the infinitely many numbers in the list, and the number we create will not be equal to any real number in the list. Which contradicts our original assumption that we listed all real numbers between 0 and 1. So the reals between 0 and 1 are not countable. And the reals are not countable. In fact the irrationals are also not countable.
We have found another infinity, much larger than
. This
larger infinity is called C, for the continuum, as the
real numbers represent all points on a line, or in the plane, or in 3-space, or
in 4-space, or in any higher order space. These sets are continuous, while sets
of cardinality
are
not (as there are gaps between the points). It turns out that aleph null to the
aleph null power is equal to C. I won't prove that here. My
recollection is that it is not difficult to prove.
+C = C, and
C = C.
Well, we have two transfinite numbers,
and
C. Is there perhaps any transfinite number less than
? It
turns out that there is not;
is the
smallest transfinite number. I won't prove that. Is there any transfinite
number between
and
C? The famous continuum hypothesis stated that there is no
transfinite number between
and
C. It has been shown that the continuum hypothesis is
undecidable. We cannot ever know if it is true or false. We will never find a
transfinite number between
and
C. We can define a kind of set theory where there is no such
transfinite number, and we can define a kind of set theory where there are such
transfinite numbers, and both kinds of set theory are consistent and valid. In
other words, we cannot prove that either set theory is the right one. And so we
don't know if C is aleph one, the next greater transfinite
number.
We can find transfinite numbers greater than C. In fact there are infinitely many different transfinite numbers. CC is a larger transfinite number. And if X = CC, then XX is an even greater transfinite number.