aleph-null

Joke Nodes: Geek Jokes: aleph-null

A set with cardinality aleph-null is a set which can be put in one-to-one correspondence with the set of natural numbers. These sets are called "countable" sets for this reason. Many important sets in mathematics are countable, such as:

• The natural numbers (N)
• The integers (Z)
• The primes (P)
• The rationals (Q)
• The algebraics (A)
• The Gaussian integers (G)
• And many many more...

One useful property of a countable set is the ability to put a non-dense ordering on it. This means that one can define "less than" such that there is no number "between" two adjacent numbers. This is very useful for certain kinds of proof, e.g. proof by induction.

As with all the transfinite numbers, aleph-null does not obey the same laws of algebra as finite numbers. Here are a few basic facts about algebra involving aleph-null:

k + aleph-null = aleph-null for any finite k

k aleph-null = aleph-null for any finite non-zero k

aleph-null^k = aleph-null for any positive k

However, k^aleph-null = aleph-one for any finite k > 1

This has been part of the Maths for the masses project

/ | 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 |
--|---|---|---|---|---|
. |
. |
. |

n6,

Cantor's Continuum Hypothesis was that 2^aleph-0=aleph-1.

What hasn't been determined is whether or not this is true, hence the title "hypothesis."

What has been determined is that granting it true we resolve one model of mathematics and granting it false we resolve another.  Both models are equally true, in a vague sense.  It helps to remember we are only dealing with a priori frameworks and not physical things (or are we is it right to pose unanswerable hypothesis when describing unanswerable hypotheses??).

At this point we are faced with an aesthetic decision whether we want a simplified model or a more elegant one.