# Universal locally countable partial order

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 $$P$$?

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