Defining Aleph-One: The Elusive Cardinal Number

[ehrenfest]

Homework Statement

aleph-0 is defined as the cardinality as the integers right?

How is aleph-one defined? Is the only definition as the next largest cardinal number after aleph-1?

Annoyingly, my set theory book (kaplansky) doesn't seem to define it anywhere. Also the wikipedia article just confused me more.

Homework Equations

The Attempt at a Solution

 

[morphism]

How much do you know about ordinal numbers?

 

[ehrenfest]

I see now. The problem was that Kaplansky uses alpha-two on page 47 but defines ordinal numbers on page 57.

 

[n_bourbaki]

Surely one of the real problems with defining aleph-1 is that what it is depends on what set theory you use. If you accept CH, then it is the cardinality of the real numbers. But it need not be. I have this idea that there may be infinitely many cardinals between aleph-0 and c (cardinality of the continuum, or 2^aleph-0) in some models, but it is just a feeling.