Given partial order $(P(\mathbb{N}), \subset)$. I have to find chain and antichain in this partiar order equipotent to $\mathbb{R}$. Actually, I don't have any idea how to begin :) Any hints?
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$|\mathbb Q|=|\mathbb N|$. – Andrés E. Caicedo Nov 18 '14 at 17:53
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Many duplicates. Chains: one, two, three. Antichains: four, five... there are probably more of each type. – Asaf Karagila Nov 18 '14 at 18:01
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HINT: $|\Bbb Q|=|\Bbb N|$, so you can replace $\Bbb N$ by $\Bbb Q$. Then for the first part think about Dedekind cuts — specifically, their downward halves, say. For the second part, take a look at this question and the answer given there (or the answers given at the linked earlier question).
Brian M. Scott
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You beat me for that one link, but I've beaten you for four others! :-) – Asaf Karagila Nov 18 '14 at 18:02