aleph-null (thing)

As 10998521 has pointed out, aleph₀ is the cardinality of both the integers and the rationals. Essentially, that means that there are just as many counting numbers as there are fractions. This can be demonstrated as follows.

Observe the following division table:

/ | 1 | 2 | 3 | 4 | 5 | ...
--|---|---|---|---|---|-----
1 | 1 |1/2|1/3|1/4|1/5|
--|---|---|---|---|---|
2 | 2 | 1 |2/3|1/2|2/5|
--|---|---|---|---|---|
3 | 3 |3/2| 1 |3/4|3/5|
--|---|---|---|---|---|
4 | 4 | 2 |4/3| 1 |4/5|
--|---|---|---|---|---|
5 | 5 |5/2|5/3|5/4| 1 |
--|---|---|---|---|---|
. |
. |
. |

It's not difficult to see that this table contains all the positive rational numbers. Now, we can set up a one to one correspondence between the cells of the table and the integers: Start by labeling the upper left cell "1". Then the cell to its right (cell (1,2)) is "2". The one to the lower left of that (cell (2,1)) is "3". The next cell is (3,1), then (2,2), and so on, zigzagging across the table and skipping anything we've seen already. Now if we imagine copying the table but changing the sign of all the fractions and using the negative integers as labels, we see that the set of rationals has the same cardinality as the set of integers, namely aleph₀.

As a side-note, the larger cardinality of the set of reals arises from the fact that there are uncountably many irrational numbers. Though I won't actually prove fact here, I will give a bit of a suggestion of why it might be true:

2^aleph-null is the cardinality of the power set of the integers. That is, the set of all unordered subsets of the integers. (See Cantor diagonalization for a justification of |R| > |N|.) Now, if we pick one set out of this power set, put it in some order (mumblemumbleaxiom of choicemumblemumble) and throw in a decimal point, we have some real number which is distinct from any other real number which could be constructed from one of the other elements of the original set.

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One quick correction: 2^aleph-null is not necessarily aleph₁. aleph₁ is, by definition, the cardinality of the smallest uncountable set. 2^aleph-null is the cardinality of the reals, commonly called c. So since |R| > |N| and |N| = aleph₀, we have c ≥ aleph₁. It has been proven that it is unprovable whether c > aleph₁ or c = aleph₁. This is known as the continuum hypothesis.