Algebra Seminar talk
2026-11-13
Asaf Karagila
The terrible truth about iterability
Abstract:
The chain condition is one of the most useful combinatorial properties to tame the method of forcing. Alas, the proof that a forcing notion with the countable chain condition (ccc) is "well-behaved" relies, ultimately, on the Axiom of Choice. With Noah Schweber, we defined a number of variations of ccc and studied them in the absence of the Axiom of Choice. Philipp Schlicht remarked in 2024 that the one we suggested actually lacks an iteration theorem. Namely, ZF does not prove that a finite support iteration of ccc forcings is itself a ccc forcing. Recently, with Constance Bromham, we formalised this remark and studied iterations of ccc forcings in ZF+DC. In the process, we uncover a choice-free proof that the iteration of proper forcing is proper.