Dabbling in Infinity - Part II
Boy this math stuff can be hard to understand. I mean concepts
like infinity, differential calculus, set theory. Good grief.
John von Neumann, the famous mathematician, said "Young man, in
mathematics, you don't understand things, you just get used to
them." That's exactly the way I feel sometimes when I myself
read and study about this most curious subject. I guess the
reason why I excel at explaining concepts in mathematics is that
I have experienced so much frustration in trying to understand
this subject from another's perspective that I go out of my way
to break things down so that even a baby could understand. Well,
not a baby, but you know what I mean. Often I go and do research
on the internet and read articles there. It is sad that well
written, intelligible articles are hard to find. And this is
coming from someone who has studied mathematics at the graduate
level!
I remember back in graduate school, how hard it was to read the
textbooks and make any sense out of what was being taught. The
stuff literally was layered with abstraction after abstraction.
The theoretical material was explained in a way that would daunt
even the most scholarly. No wonder people never really get past
basic mathematics, and no wonder this subject is easily despised
by so many. Goodness gracious, even mathematicians find the
material mind boggling. However, one can come to understand the
material if--and only if--it is broken down in a way that humans
can comprehend. Granted, mathematics is hard. Especially when
one starts to delve into concepts like infinity and the like.
Given this, I still say that if you have the right teacher to
explain something, progress can be made in this subject. It is
indeed true that in mathematics particularly, the teacher can
make you or break you. Thus you should not feel too bad and
think too little of yourself if you had trouble with mathematics
in school.
Now let's get back to this concept of infinity and the case
made that this is not a universal concept. What is meant by this
is that there is not a unique infinity. There are actually--and
this can be shown mathematically--infinitely many kinds of
infinity. (Wow, if this is not an amazing fact then I don't know
what is.) To get you to understand this, ultimately I am going
to show that the set of numbers {1, 2, 3, ...}, which go on
forever, cannot be paired with the decimal numbers between the
interval 0 and 1.
Let's examine this a little more carefully. If I take a set of
objects, let us say, the set of odd numbers between 1 and 9,
then I have the set {3, 5, 7}; a set consisting of three
members. If I now take the set of three boys named {John, Jules,
and Steven}, I have a set of three boys. How can we determine
which set has more elements? Well, it is quite intuitive that we
can pair off the elements from each set and we see that no
element from the set {3, 5, 7} would be unpaired with an element
from the set of three boys. Of course, there are a couple of
ways we could do the pairing but the main thing is that each set
has the same number of elements, although these elements are not
necessarily the same thing. What we are interested in is that
the number of elements of each set is the same and is equal to
three. We call three the cardinality of each set. Since the
number three is finite, the cardinality of these sets is finite.
Clearly for the set of counting numbers {1, 2, 3, ...}, we can
not associate a number which represents its cardinality. If we
chose a billion to represent the cardinality, this would be
wrong because the set has elements which surpass one billion.
Similarly if we chose any predetermined number, regardless of
its size, we can always go out further in this set and find a
bigger number. Mathematicians call the cardinality of the set of
counting numbers "aleph nought," which is the Hebrew letter
aleph with the subscript of 0. This turns out to be the first
"transfinite" number, or number which is not finite. Some very
strange and bizarre things happen when we enter the realm of
transfinite numbers. For one, it seems as though certain
infinite sets have more elements than another infinite set, when
in actuality both sets have the same number of elements! What?
How can this be? Follow this example carefully. Take the
counting numbers {1, 2 ,3...} and the set of odd counting
numbers {1, 3, 5...}. It would seem that there should be more
counting numbers than odd counting numbers, but in fact, this is
not the case. How can we show this?
The technique of pairing elements is the way we show that two
different infinite sets have the same cardinality. This pairing
is called a "one-to-one" correspondence. For the two finite sets
mentioned above it works like this: we pair 1 with John, 2 with
Jules, and 3 with Steven. Therefore each element of the first
set is paired with exactly one element from the second set.
There is no spillage, or leftover elements that are not paired.
That is all there is to the notion of one-to-one correspondence.
It makes sense then if we can pair each element of one infinite
set with each element of another infinite set so that there is
no spillage, then the two sets, though infinite, must have the
same number of elements, or cardinality. To show how this is
done with the counting numbers and odd counting numbers is very
simple: pair 1 with 1, 2 with 3, 3 with 5, 4 with 7, 5 with 9,
and so on, ad infinitum. Since the odd numbers go on forever, we
can go further out in this set to pair the elements from the
counting numbers. Thus even though at first it seems that the
set {1, 2, 3,...} is more numerous than the set {1, 3, 5, ...},
this we now see is entirely untrue. Strange realities indeed
exist when it comes to infinite sets!
In Part III, I will show you how and why the decimal numbers
between the interval 0 and 1 are more numerous--that is display
an infinity which is greater--than the set of counting numbers,
though the latter go on forever. Very strange indeed! What
bizarre things lurk in this world of transfinite numbers.