Spencer talked today about weakenings of Martin's Axiom of the form "All c.c.c. posets have a nice property." The nice properties were things like "$\omega_1$-Knaster", or "$\sigma$-centered." Actually, Todorcevic--Velickovic proved that $MA_{\aleph_1}$ is equivalent to "all c.c.c. posets are $\sigma$-centered". We saw how Martin's Axiom implies these other principles.
The main thing Spencer talked about was that "All c.c.c. posets $\mathbb{P}$ have $\mathbb{P}\times\mathbb{P}$ c.c.c" implies that the continuum hypothesis fails. To prove this, he uses the CH to construct an entangled linear order, and shows that the poset of chains of this entangled linear order is c.c.c. but its square is not. The proof was very involved, and I'll try to write up some more about this when I have some time to think about the main underlying ideas. I remember reading about entangled linear orders from Shelah's book on cardinal arithmetic, but not being very motivated by it. This talk showed me the power of these linear orders when proving that the poset of chains is c.c.c.
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