cardinal number

Finite cardinal numbers:

0 = The number of elements in the empty set: {}

1 = For example, the number of elements in the set of subsets of the empty set: { {} }

2 = Examples: The number of elements in the set of subsets of a one-element set. Or, the number of elements in this set: { {}, { {} } } = the set which contains the empty set along with the set containing the empty set.

etc....

Infinite cardinal numbers:

Aleph-0 = "Countable" = the number of elements in the set of finite cardinal numbers = the number of elements in the set of natural numbers { 1, 2, 3, 4, .... } = the number of integers = the number of rational numbers = the number of algebraic numbers = the number of lattice points on a grid .....

Aleph-1 = The smallest cardinal number greater than Aleph-0.

The number of subsets of a countable set = the number of real numbers = the number of transcendental numbers = the number of irrational numbers = the number of points in a line or plane...... The continuum problem of whether this is in fact equal to Aleph-1 is unprovable using the usual ZFC axioms of set theory. Either the continuum hypothesis (the statement that they ARE equal) or its negation can therefore be used as additional axioms.

There are infinite infinite cardinal numbers since a greater infinity can always be generated by taking the cardinal number of the set of subsets (or power set) of a set with a lesser infinity of elements.

0 isn't really there at all. It's just the space where it would be. Don't fall in the hole. We put up signs, once, in Arabic, but they fell in too.

1 is a patch job; the smallest natural number you can stick in to make a difference, bridge a gap, save your place. Essential, yes, but as common as dirt, it shows up everywhere.

2 is slow but strong. Dependable. It never shows up in binary but we all know the system would collapse without it. You can go a long way with the powers of two.

3 is magical. A little fairy with twisting tumbling powers you couldn't imagine doing you any harm. But they might. Things rarely come out even when 3 plays its hand.

4 is square, reliable, used in lots of foundations as 2's older brother, but it has been spotted with 3 about a dozen times. What was he up to? Death in Japan.

5 is the white wizard. It came up with the decimal system with inspiration from 2, and on its own it can get you from 0 to 100 in twenty steps. Many fives make light work.

6 is a 3-D artist we don't entirely trust. 3 put a hex on Six a long time ago and made him think he was a perfect number. Now he turns up a lot when she's working and just stands there repeating himself.

7 is thought to be magical and perfect - he turns up in all the stories - but he's really just lucky. Seven is oldest of the primes and hard to talk to as a result.

8 is the older sister of 2; she never had any children but she's a kindly if eccentric witch who treats 2 like her grandson.

9 stays up in his tower, mostly, visited only by 3. Sometimes he comes down and performs party tricks for the other numbers on their birthdays, but otherwise Nine's up in his tower, looking into the abyss imagining larger digits in other bases and higher primes spiralling up and away just out of his reach.

Real poets kick ass: The node Numbers contains a Robert Creeley poem in a similar vein.