#### 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

Homework Helper
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

Homework Helper
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.

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