Quick question about cardinals

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The discussion revolves around defining uncountable sets of cardinal numbers, questioning how such sets can exist if every set of cardinals "up to x" appears countable. It highlights the distinction between uncountable ordinals and cardinals, suggesting that uncountable ordinals can define uncountable cardinals. The conversation challenges the assumption that all cardinals are countable and calls for a proof to support this claim. The mention of uncountable ordinals, like w, emphasizes their role in establishing uncountable cardinals. Ultimately, the dialogue seeks clarity on the relationship between cardinals and ordinals in set theory.
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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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