Mike Raiford wrote:

> What is Aleph-null? Is it the set of all integers? or is it something a 
> little different. I know it's basically a different sort of infinity...
> 
> [A Quick wiki detour later] Oh, Aleph-Null is basically any infinite 
> set, Aleph-One would be a set of all ordinals (positive integers and 0) 
> .. Interesting

Aleph-null is the *size* of a set (specifically, the set of natural 
numbers). The technical term is "cardinality".

The set of all positive numbers (including or excluding zero) is 
Aleph-null. In fact,

   Aleph0 + x = Aleph0
   Aleph0 * x = Aleph0
   Aleph0 ^ x = Aleph0

assuming that x < Aleph0 (i.e., x is finite). For this reason, the set 
of all integers (positive and negative) has size 2 * Aleph0 = Aleph0. In 
other words, the set of all integers is THE SAME SIZE as the set of 
positive integers. (So it really isn't especially important exactly 
which set you use as your definition.)

Additionally, the set of all 2D coordinates has cardinality Aleph0 * 
Aleph0 = Aleph0, so that's the same size too. The set of all rational 
numbers also has the same size, as does the set of all algebraic numbers 
(i.e., roots of polynomials - so that includes irrational square roots 
and the like).

However, the set of all *real* numbers includes also transcendental 
numbers - numbers which are not the root of any polynomial. And *this* 
set has cardinallity Beth-one. And Beth-one > Aleph-null.

   Aleph-one = 2 ^ Aleph-null

(Note that Aleph-null ^ 2 = Aleph-null, which isn't the same thing at all!)

If the continuum hypothesis is true then Beth-one = Aleph-one.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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