Google Hilbert's Hotel (or, Hilbert's Paradox) and a number of good descriptions are bound to come up so I won't bother here with it. But I would like to talk about another form of infinity that really captured my imagination. I forget now where exactly I read of it but I'm sure it's in my collection of books on mathematics. It goes something like this:
The irrationality of the square root of two is a point somewhere in the number line. Its approximate value is 1.41421, but the decimal expansion is without end and no apparent pattern will ever emerge no matter how far into the infinite sequence one gets. I spoke recently of the hapless Hippasus who was thrown overboard for having discovered the irrationality of the number.
Anyhoo, the diagonal of a unit square is where the number comes from, and following Pythagoras' famous equation it looks like this:
If you line up the diagonal to the number line it'll land somewhere roughly halfway between 1 and 2. But not quite. Divide up the interval between 1 and 2 into ten equal parts (microscope 10x) and see if lines up...nope. Divide up the interval to 100 parts (microscope 100x)...still nope. Divide up into thousand equal parts (microscope 1000x)...nope. How about a billion equal parts...still it wouldn't line up. One can keep repeating the division of the interval forever but the length of the square root of two will never line up exactly.
Strange as it is to our rational minds (no pun intended) there are intervals along the number line that can never be lined up exactly to equal division of parts. The point is there, certainly. But why it's called an "irrational" number is by its very definition not possible to divide into whole number ratios.
It is said that Hilbert's Hotel can accommodate an infinite number of occupants, that even if an infinite number of tour buses with infinite number of tourists arrived one night the hotel would not run out of rooms to let. The arithmetic relies on the notion of one-to-one correspondence between one type of number to another type (square numbers to whole numbers, odd numbers to even numbers, etc.).
The arithmetic of the infinite (the set theory invented by Georg Cantor that spawned it) is a subtle form of arithmetic but it does not break nor violate any logical argument or any derivable output of the axioms of number theory. It proves the very concept of an irrational number (by way of the Dedekind cut) and lays the foundations of calculus on a rigorous footing by defining the concept of the limit among other things.
It is counter-intuitive but utterly above rational reproach (ie, the arguments are derived from sound and demonstrable logic). It transcends the very fabric of maths and yet at the same time births and sustains all that exists as number.
In contemplating this stuff I feel that I'm at once humbled and elevated to touch the face of G*d. Truly, there are limits to human knowledge and what we may come to know but there is a great, undefinable comfort in knowing that we exist in an infinite ocean of being.