In continuation of my discussion on infinity and its
implications with the divine, I should mention that the concept
of there existing infinities beyond imagination is quite
difficult to comprehend. If you read my poem, "How Can this Be?
you read in verse the proof that shows clearly that there is no
such thing as one kind of infinity. (See my ezine article "How
Can this Be") The extension of this most curious fact is that
there are actually an infinite number of infinities!
Occasionally I wax metaphysical in conversations with my
uncle and the other evening we were discussing some points
regarding the spiritual realms. In passing, I brought up the
topic of infinity and I asked him his impression of it. His
response, which is typical of most people, is that infinity is
just that--infinity: something that never ends. But how do we
make this vague notion somewhat more concrete? I pointed my
uncle's attention to the set of natural, or counting numbers.
This set comprises the familiar numbers {1, 2, 3, ...}. The
numbers go on and on, falling like dominoes, and never reaching
a "biggest one." This process is easy to grasp and presents no
ordinary difficulty for the average person. What does become
difficult to understand is why the infinity typified by this set
of numbers is not unique.
Now let's delve a little more deeply into this curious set of
numbers and in the topic of infinity in general. This set of
counting numbers obviously never ends. If you have ever seen a
chronometer counting hundredths of a second, then you have seen
how fast the digits representing the hundredths of a second whiz
by, not appearing for any length of time sufficient to allow
recognition of the appropriate digit. And this is for hundredths
of a second. Imagine the same chronometer counting off
thousandths of a second. Now imagine this going on from, let us
say, ten thousand years ago and continuing for another ten
thousand years, starting with 1 and such that each thousandth of
a second would represent the next sequential counting number.
Think of how far along in the set of counting numbers you would
be. We could actually compute the number but we are only
interested in trying to conceptualize how large potential
infinity could be. Now that we have this huge number in hand, we
could do whatever we wanted with it to project ourselves much
further out in the set of counting numbers. We could multiply it
by itself ten times (the mathematical way of saying we can raise
the number to the tenth power); we could multiply it by itself a
hundred times, a thousand times, and so on. We could then take
the largest product and do the same process all over again. How
big is this set?
This set is so large--never ending in fact--that we should be
able to use it to compare to anything else that is infinite,
right? Wrong. And in a continuing article on this most
fascinating subject, I will discuss how this notion of one
universal infinity is completely wrong. Thus, if sets of numbers
can shatter our preconceived notions of a concept like infinity,
which is more or less universally accepted as something that is
real, what more can we uncover by plunging into the mysteries of
numbers and mathematics in general? Stay tuned......
About the author:
Joe is a prolific writer of self-help and educational material.
Under the penname, JC Page, Joe authored the classic of
mathematical ABC's Arithmetic Magic. Joe is also author of the
charmingly pithy and popular ebook, Make a Good Impression Every
Time: The Secret to Instant Popularity; the seminal collection
of verse, Poems for the Mathematically Insecure. For more
information, visit his website at
www.mathbyjoe.com.