How to define an uncountable set of cardinal numbers

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