# How is aleph-one defined?

1. Jul 24, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution

2. Jul 24, 2008

### morphism

Re: aleph-one

How much do you know about ordinal numbers?

3. Jul 24, 2008

### ehrenfest

Re: aleph-one

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

4. Jul 25, 2008

### n_bourbaki

Re: aleph-one

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.