INNER MODEL THEORY AND LARGE CARDINALS

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Research in this project was devoted to inner model theory, large cardinals, forcing theory, cardinal invariants of the continuum and other subfields of set theory, as well as to the interplay between them and to applications to other areas of pure mathematics. We briefly sketch the main topics and results.1.Maximality properties of inner models and elementary embeddings. For example, we investigated under which circumstances the Jonsson property holds for a given cardinal in the core model K if it holds in the universe V.2.Research on infinite time Turning machines. We showed in particular that, if λ is the supremum of the writable ordinals, i.e. ordinals which arise at outputs of computations on input 0, and γ is the supremum of clockable ordinals, i.e. ordinals which are lengths of halting computations on input 0, then λ=γ.3.We proved the set of reals Cohen over a model of ZFC must either be empty or non-meager.4.The effect of cardinal invariants of the continuum on combinatorial properties of uncountable cardinals. For example, we proved that additivity of the null ideal implies that Martin's axiom MA holds for any Cohen algebra. On the other hand, we showed it is consistent that the continuum c is large and covering of the null ideal as well as the combinatorial principle * hold.5.Say that PSP(κ, Γ) holds if every set in the pointclass Γ of size at least κ has a perfect subset. We showed that PSP(ΝィイD21ィエD2, GィイD2ΝィエD2ィイD21ィエD2) holds if and only if σ 【greater than or equal】 ΝィイD22ィエD2 where GィイD2ΝィエD2ィイD21ィエD2 is the class of sets which are intersections of (at most) ΝィイD21ィエD2 many open sets and σ is the dominating number.6.We proved that every maximal confinitary subgroup of Sym(ω), the permutation group of the natural numbers, has size at least the cardinality of the smallest non-meager set reals.7.Completing a cycle of results initiated by Shelah and Spinas, we obtained that if c=ΝィイD22ィエD2 then there is a Gross space over every countably infinite field.