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Call a poset locally countable if the set of predecessors of every member of the poset is countable. Is the following consistent?

There is no locally countable poset $P$ of size continuum such that every locally countable poset of size continuum embeds into $P$?

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    $\begingroup$ Out of curiosity, is the negation of this (i.e., the affirmative: there exists such universal $P$) consistent? $\endgroup$ – YCor May 28 at 15:58
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    $\begingroup$ Yes. Under CH, the Turing degrees witness this. This is due to Sacks. $\endgroup$ – Ashutosh May 28 at 16:20
  • $\begingroup$ Do you want them to embed as initial segments? $\endgroup$ – Joel David Hamkins May 28 at 20:27

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