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Seminar on Reckoning

The grounded Martin's axiom

Miha Habič
FF UK

 

Seminar on Reckoning
Wednesday, 13. September 2017 - 11:00 to 15:00

Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building


We will examine the notion of a grounded forcing axiom, which asserts that the universe is a forcing extension by a forcing notion from a particular class and that the usual forcing axiom holds for forcings from that class coming from the ground model of the extension. We shall focus in particular on the grounded Martin's axiom, where the universe is a ccc extension. The principle has some of the combinatorial strength of MA, but allows for more flexibility (for example, a singular continuum). Furthermore, it is more robust under mild forcing extensions than full MA, since it is often preserved after adding a Cohen or a random real. We will also briefly glance at grounded versions of other forcing axioms, such as grounded PFA, and outline some open questions in the area.