1. Aug 22, 2007

### Dragonfall

Assuming we define the cardinal number for a set A as the least ordinal number b such that A and b are equipollent, how would you define an uncountable set of cardinal numbers?

2. Aug 22, 2007

### matt grime

It's just an uncountable set. That it happens to be a set of cardinals is immaterial.

3. Aug 22, 2007

### Dragonfall

What I mean is that under this model every set of cardinals "up to x" seems to be countable. So how to define an uncountable one?

4. Aug 22, 2007

### matt grime

How does that imply that all cardinals are countable? Why not post a proof of that statement if it 'seems' to be so. Hint, let w be an uncountable ordinal. Such exist. It is not in bijection with any countable initial segment, so it must define an uncountable cardinal too.

Last edited: Aug 22, 2007