Quick question about cardinals

[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?

 

[matt grime]

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

 

[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?

 

[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.