Take charge young man and reckon your soul,

With math it’s possible, so why contest?

Why close the door to wonders untold?

Dimensions unseen, or places blessed?

Did you ever wonder about the things you could do,

Simply with numbers? You can do miracles—it’s true!

The whole numbers, for example. They never end,

Yet contain an infinitude of primes, while seeming to blend,

Composites together, perfect squares and like kin,

In such perfect harmony within a land of no sin.

And as infinite as they are, still they cannot contend,

With the more numerous reals, which further extend,

To infinities beyond what we normally comprehend.

When Cantor first discovered this, he blushed with fear,

At the import and magnitude of this extraordinary idea.

Oh what wonder, oh what a day!

But how could this be you say?

In fact, pay heed and listen, it’s true,

In such perfect harmony within a land of no sin.

Read More

There are more real numbers between 0 and 1,

Then all the counting numbers which forever grow,

And whether we move along the line fast or slow,

No matter. The reals always win and beat them out,

They outnumber and defeat, exercising their clout,

Oh what a wonder, oh what a day!

But how could this be you say?

Even more amazing and I will put this forth,

Is that we can shrink the interval between 0 and 1,

To anything you like. Now wouldn’t that be fun?

For example, take 0.1. I assert—and I do this with care,

That there are more reals between 0 and 0.1,

Then all the counting numbers, be patient don’t fear.

For I will get to the proof, but give me some time,

As I lay out some facts, and work on my rhyme.

Oh such wonders, what a glorious day!

But how could this be you say?

By extension you see, once the proof is laid bare,

You will realize strange things, very strange things indeed.

The intervals can shrink, forever vanishing,

To virtually nothing, yet listen, take heed.

The reals will still beat, within this small world,

The counting numbers, though they continue forever.

A very strange world, I am not this clever,

To ponder, to believe such, to muse to endeavor,

To enter this realm of abstract thought.

Yet do it I must, for it’s a marvelous day!

But how could this be you say?

To begin with take the counting numbers which begin 1,2,3...

They topple like dominoes going on to the next,

We show how we pair each one with a mate,

A number from the interval between 0 and 1,

And we do this in such a way that every one has a date.

It’s so simple you see, this is sure to amaze,

How this proof rings so clear, requiring only a gaze,

Of thought pure and true, let there be no delays.

Let us put forth its case, presenting it here,

For its mere truth will assuage any latent fear,

Mind-boggling, astounding, let us bless this day,

But how could this be you say?

This pairing we give a very special name,

A one-to-one correspondence, and this serves to tame,

The pairing of numbers, each real to each natural,

In such a way that the proof comes to light.

Thus we proceed, commencing with 1,

We pair it with 0.1 and we’re done,

Onto natural 2, and for this we make haste,

To pair it with 0.11, not to waste,

Any room in the middle for unwanted space,

Thus onward we go, adding 1 to the next place.

In this manner we take 3, and find him quite right,

Cavorting with 0.111 in the night.

We continue in like manner, forever and apace,

In order to finish at last this grand race.

Never realized such dealings, before this great day,

But how could this be you say?

Thus all numbers paired, and enjoying their dates,

When something would happen to alter their fates.

To unsuspecting minds, this would never occur,

But happen it did and left quite a stir.

Suddenly from afar came these numbers in tow,

Reals—whose decimal extensions would not go,

With any of the naturals because their sequence defied,

What was already established, in the correspondence we tried.

For example, no date for 0.22 could be found,

Nor for 0.55 or even 0.1333.

The reason was simple, all the naturals were bound,

To the reals already mentioned, as you can clearly see.

Thus these reals, feeling left out, decided the party to crash,

And bump off a natural, and start quite a clash.

When the naturals espied what was forming in place,

They ran with their dates in hope to save face.

Thus the infinitude of naturals just barely got away,

From the fierce unpaired reals who came to ruin their day.

What a bizarre occurrence, on such a very strange day,

But how could this be you say?

So you see there are more reals embedded in 0 to 1,

Then all the naturals though they continue to run.

The story just told relates this most curious fact,

Of how this could be, now no need to retract.

And what this implies without any reason to doubt,

Is that many infinities exist, and for this we should shout.

Thus new truths come to light in a most interesting way,

A very important lesson on this most fascinating day,

Something to ponder and to another relay,

So that progress in learning continues away.

What a most unusual tale told on this most phenomenal day,

Now at last we can say why these things are this way! QED

There are more real numbers between 0 and 1,

Read More