Orders of Infinity

I picked up this rather neat logical 'trick' from a book called 'Theories of Everything' by John D. Barrow. Any person of reasonable intelligence would, without specialist knowledge, surely think that 'infinity' is a single concept. I know I did. Take, as an example, the set of whole numbers. You could begin writing them down and never stop... onwards to infinity.

Now take the set of 'irrational' numbers (that's all numbers that cannot be expressed as a ratio of whole numbers). Intuitively, you might say that there are an infinite number of such numbers between 0 and 1, therefore there must be more of them than whole numbers, but then the concept of infinity comes swinging back in and you think, "Well, infinity is infinity, so it's all the same." But, your intuition was correct! Here's the 'proof':

Imagine writing down ALL these irrational numbers (impossible, but stay with me here). Here is what a small section looks like, all completely random:

...
0.125486312856...
0.956476213163...
0.614585484764...
0.437644646468...
0.316546416485...
0.731515565855...
0.265475487964...
0.632995245513...
...

Now take a number formed by taking a diagonal down through all those (1st digit from 1st number, 2nd digit from 2nd number... etc.):

0.15464544...

Now add 1 to all the digits:

0.26575655...

This number is different from every single number written down because it must differ in at least one digit from each (by the addition of that 1). Hence, you thought you wrote down every number possible, but there are clearly many more. Irrational numbers are said to possess a higher order of infinity than whole numbers. Expressed more poetically, does infinity require, for its expression, an infinite number of grains of sand, or, as Blake suggested, can you

"... see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour."?