Grigor Sargsyan
Grigor Sargsyan: The exact strength of Sealing
03/03/21 17:25
Tuesday, March 9, 2021 17:00
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Grigor Sargsyan (Rutgers & IMPAN)
Title: The exact strength of Sealing
Abstract: Shoenfield's celebrated absoluteness theorem says that no \(\Sigma^1_2\) fact \(\phi\) can be shown to be independent of the axioms of ZFC via the method of forcing. A set of reals is universally Baire if its continuous preimages have the Baire property in all topological spaces. Can there be independence results about such sets?
Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).
Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding
\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)
Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.
It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem.
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Grigor Sargsyan (Rutgers & IMPAN)
Title: The exact strength of Sealing
Abstract: Shoenfield's celebrated absoluteness theorem says that no \(\Sigma^1_2\) fact \(\phi\) can be shown to be independent of the axioms of ZFC via the method of forcing. A set of reals is universally Baire if its continuous preimages have the Baire property in all topological spaces. Can there be independence results about such sets?
Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).
Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding
\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)
Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.
It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem.