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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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  • $\begingroup$ $|\mathbb Q|=|\mathbb N|$. $\endgroup$ Nov 18, 2014 at 17:53
  • $\begingroup$ Many duplicates. Chains: one, two, three. Antichains: four, five... there are probably more of each type. $\endgroup$
    – Asaf Karagila
    Nov 18, 2014 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).

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  • $\begingroup$ You beat me for that one link, but I've beaten you for four others! :-) $\endgroup$
    – Asaf Karagila
    Nov 18, 2014 at 18:02

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