Hello friends, I have an idea of how to do this, but a nudge in the right direction wouldn't hurt!
Let be a class of countably infinite sets where is infinite (countably or uncountably). Prove that
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If is countably infinite this follows easily, otherwise I feel as though I can use Zorn's lemma.
That looks great, except I don't see why it follows so easily that ? I think what we could do to prove it is to take to be a sufficiently large family (maybe )and transform it into a poset by letting be a poset with . Then if we can prove that for all I think we can apply Zorn's lemma to show that has a maximal element and this is forced to be . This may very wrong, I just can't think straight right now.