There is a boy whose name's TJ,
who likes with Lego Star Wars play.
He asked his Dad to count one day,
to infinity, or at least part way.
one, two, three, four, five and higher,
he counted far until he tired.
If I give you sets A and B,
and ask if they're same-sized, let's see.
I can line them up one by one,
matching members A to B, it's fun!
If I come at end of day,
match them all and alone none stay.
Then I can say that the two sets
A and B have same size, you bets.
An infinity that's countable,
is a set S whose members all,
and be matched up in the same way,
as the set B and the set A,
Don't match S with a finite set,
but match with 1, 2, 3, and rest.
Suppose set S is countable,
match a member of S to one,
and then another one to two,
the next one of S match to three,
all the way to infinity.
If there are none left of S,
then S countable infinite is.
So let us see, boy TJ said,
if I've got it all in my head.
The set of all even numbers?
Countably infinite, I figures.
'Cause I can say that 2 is first,
and 4 is second, 6 is third,
and so on and on I can goes,
counting all evens with my toes.
The number 2 times x in S
is matched to counting number x.
You got it, TJ, mathy boy!
For you all numbers are a joy.
Now try another kind of set.
This one's infinite and yet,
it can't be counted or matched up,
to the natural numbers, nope.
A set like that a math wiz calls
infinitely uncountables.
The real numbers are one such set,
you can't count them I just bet.
Which is first, zero point one?
but what about point zero one?
The way to prove a set like that,
can't be numbered by any cat
(without or with a funny hat),
the proof is quite a digestion
the process of explaination:
"By diagonalization."
Alas my poem cannot cope
with proofs like that, not any hope.
Suffice to say it may surprise
that some large sets look to your eyes
uncountably infinite in size.
Yet you can count them if you tries.
Rational numbers for example,
are infinite yet countable.
Just ask me fine and I'll explain,
how one can count a set so fine.
But poem's not the place to say
a proof like that to my TJ.
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