Set theory provides a (hopefully) stable foundation to mathematics and is itself a sophisticated area of mathematics. In addition, methods from set theory are useful in various other areas (topology, functional analysis, group theory, etc.). Our work focuses on the interplay between forcing axioms, the extent of the forcing method, large cardinal axioms, infinite combinatorics, and issues of definability, and deals with the relationship between various extensions of the standard axiomatization of set theory (ZFC). Some specific problems proposed in this project pertain to the combinatorial structure of H(omega_3) and of higher fragments of the set-theoretic universe under high forcing axioms, an area relatively unexplored until recently but which seems to be blooming now. Other problems concern the extent to which very strong forcing axioms decide, in the presence of large cardinals, the theory of H(omega_2) modulo forcing.

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