CARDINAL INVARIANTS OF THE CONTINUUM AND THEIR APPLICATIONS

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Research in this project was devoted to the interplay between cardinal invariants of the continuum and iterated forcing theory, as well as to applications of both areas to algebra. We briefly sketch the main topics and results.1. Iteration along a template. Using an axiomatic approach to Shelah's recent technique of iteration along a template which was originally developed to show the consistency of σ < a, we obtained several new results. For example, we proved it is consistent that a_g>max{a,non(M)} where a_g denotes the size of the smallest maximal cofinitary group. We also showed that the almost disjointness number a can consistently be a singular cardinal of countable cofinality.2. Shattered iteration. Using a sophisticated new iteration technique built from a system of compete Boolean algebras which add both Cohen and random reals, we proved the simultaneous consistency of cov(M)=non(N)=N_2 and cov(N)=non(M)=c=N_3.3. Perfect set axioms. Say that PSP(k,Γ) holds if every set in the pointclass Γ of size at least k has a perfect subset. Let G_N_1 be the class of sets which are intersections of (at most) N_1many open sets. We showed that after iteratively adding N_2 Sacks reals over a model for CH, P5P(N_2, G_N_1) holds.4. Cardinal invariants related to evasion and prediction. We proved in ZFC that b【less than or equal】b_2 where b is the unbounding number and b_2 is the constant prediction number, thus answering a question of Kamo. In joint work with Shelah, we obtained several related results, for example that the k-constant prediction numbers for different k can consistently be different.5. The cofinality of Sym(ω). The cofinality c(Sym(ω)) of the symmetric group Sym(ω) is the least cardinal k such that Sym(ω) can be written as the union of a strictly increasing chain of proper subgroups of length k. In joint work with Losada, we proved that g【less than or equal】c(Sym(ω)) holds in ZFC, thus partially answering a question addressed by Thomas.