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forcing; axiom of choice; non-AC forcing; ZF
Typically, set theorists reason about forcing constructions in the context of Zermelo--Fraenkel set theory (ZFC). We show that without the axiom of choice (AC), several simple properties of forcing posets fail to hold, one of which answers Miller's question from the work: Arnold W. Miller, {Long Borel hierarchies}, MLQ Math. Log. Q. {54} (2008), no. 3, 307--322.
[1] Karagila A.: Do choice principles in all generic extensions imply AC in $V$?. Answer to a MathOverflow question, 2018.
[2] Kunen K.: Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1983. MR 0756630 | Zbl 0534.03026
[3] Miller A. W.: Long Borel hierarchies. MLQ Math. Log. Q. 54 (2008), no. 3, 307–322. DOI 10.1002/malq.200710044 | MR 2417803
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