Choiceless chain conditions

Abstract

Chain conditions are one of the major tools used in the theory of forcing. We say that a partial order has the countable chain condition if every antichain (in the sense of forcing) is countable. Without the axiom of choice antichains tend to be of little use, for various reasons, and in this short note we study a number of conditions which in $$\mathsf {ZFC}$$ ZFC are equivalent to the countable chain condition.

Assessing the influence of the axiom of choices variations on mathematical foundations and subdisciplines

“…(1) pinpointing the strength of axioms and examining their relationship with large cardinals (one example that was given by Woodin is that extremely large cardinals are at some level, trying to "mimic" the axiom of choice) and ( 2) use axioms like "The existence of … cardinal" (often referred as large cardinal axioms) to replace common axioms for proving or constructing things (like the Grothendick's universe) [30]. The adoption of the axiom of constructability would mean giving up on such nice tools, though there has been attempt to bolster this axiom so that the model could hold large cardinals (like Woodin's Ultimate-L program) [31]. It is also worth known that some mathematicians think that the fact that V=L provides definite answers to certain questions also, on some level, damaged the freedom of the set universe, and they often believed in the multiverse approach [32,33].…”