DIALOGUE BETWEEN MACHINE AND MAN
Hey man! Do you have time, would you agree,
To chat about real numbers now with me?
My friend, you've never seen a real real number.
You cut them off before they're halfway done.
Nor twenty decimal places, nor a thousand,
Are adequate to hold a single one.
I think you underestimate me, man.
My software package is the best in town.
See here's a little program that computes
The first n places of the number pi.
You choose the n, and if you have the time
You'll get as much of pi as you can stomach.
Such strings of digits are my cup of tea,
I mind not that they go on endlessly.
You've barely scratched the surface none the less.
The set of numbers you can calculate
With programs, like the one you wrote for pi,
Can be enumerated one by one,
And Cantor showed that, given such a list,
There is at least one number that is missed.
Insanity! What number has been seen
In all the world that I can't calculate?
Nor can you list the programs that compute
Each digit in a number's decimal string.
A child of ten can write the code to list
The programs my compiler will accept.
But here the programs must consist of those
That endless strings of digits do produce.
Write such a code, man, and the pigs will fly.
There's no such code. Look here, I'll show you why.
I know why, Mac, you just use Cantor's proof
To show that there is no recursive list
That itemizes all recursive functions.
The problem is that every list you know
Is general recursive---that's your world.
It's a paradox you'll never understand,
That I can count your numbers one by one
Despite your proof that it cannot be done.