Friday, April 17, 201510:00 amSet theory seminarGC 6417

# The boldface resurrection axioms

## Thomas Johnstone

### The New York City College of Technology (CityTech), CUNY

A few years ago, Joel Hamkins and I introduced a new class of forcing axioms, the Resurrection Axioms for various classes Gamma of forcing notions. The point of resurrection is that statements that are true in H_c (the collection of sets with hereditary size less than the continuum), but whose truth has been destroyed by some forcing notion in Gamma, can be resurrected by some further forcing (also in Gamma, if desired), i.e. the truth of the statements can be forced to hold again in H_c of some forcing extension. We investigated the resurrection axioms for various classes Gamma of forcing notions, such as c.c.c. forcing, proper forcing, or all forcing.

In this talk, I will discuss the Boldface Resurrection Axioms, which generalize the usual resurrection axioms by allowing predicates for subsets of the continuum c in the statements that are true about H_c, but whose truth has been destroyed by some forcing in Gamma and that need to be resurrected by further forcing. We shall show that various instances of the boldface Resurrection Axioms (such as for the classes of c.c.c. forcing or proper forcing) are equiconsistent with the strongly uplifting cardinals, a new large cardinal notion. This is joint work with Joel Hamkins.

Thomas Johnstone is a professor of mathematics at the New York City College of Technology (City Tech), CUNY. His research in Set Theory includes large cardinals, forcing, and indestructibility.