Forcing with side conditions and forcing axioms

This is a project on set theory. Set theory is an extremely general mathematical theory that provides a solid foundation to virtually all of contemporary mathematics, and it is itself a deep area of mathematics. It is embodied by the first order theory ZFC (Zermelo Fraenkel set theory with the Axiom of Choice), often supplemented with additional axioms (for example, axioms asserting the existence of `very large` cardinals, forcing axioms, determinacy axioms, acioms phrased in terms of cardinal arithmetic, or axioms asserting the existence of objects with interesting combinatorial properties). The main topics of this project are forcing with symmetric systems of structures as side conditions and forcing axioms. There will be be an additional complementary topic that will go under the title set theory with restricted choice and very strong large cardinal hypotheses. The method of forcing with symmetric systems of structures as side conditions, recently developed by myself, in collaboration with my former student M.A. Mota, provides an alternative approach to the construction of models of set theory via iterated forcing. With this method we have for example built models of interesting fragments of classsical forcing axioms in which the size of the continuum is large (classical forcing axioms, which are natural principles asserting a certain maximality of the universe with respect to its forcing extensions, tend to imply that the size of the continuum is exactly the second uncountable cardinal). We have also built models of high analogs of the Proper Forcing Axiom that had not been considered before. One of the main goals of this project will be to explore further applications and extensions of this method. For example I will work in the construction of models of the Continuum Hypothesis (CH) with interesting properties, and of models with high superatomic Boolean algebras of small width, and I will also work on proving the consistency of high analogs of the Proper Forcing Axiom and will develop the theory provided by these axioms. Another - related - goal will be to show that the forcing axiom for a natural subclass of the class of proper posets implies that the size of the continuum is exactly the second uncountable cardinal. This will involve developing new ways of coding real numbers by ordinals in the context of forcing axioms. Another goal of this project regarding forcing axioms concerns the exact relationship between a certain maximal principle of generic absoluteness defined by Woodin and classical forcing axioms, and yet another one concerns the possibility of proving that Jensen`s diamond principle provides, modulo large cardinals, a maximal theory, relative to forcing, for the relevant initial segment of the universe. A solution to this problem would complement the result - obtained in 2009 by myself, together with J. Moore and P. Larson - that there is no corresponding maximal model for CH. In the additional topic set theory with restricted choice and very strong large cardinal hypotheses I will explore a relatively new line of research consisting in the derivation of combinatorial properties of the universe in the absence of the Axiom of Choice or assuming just very limited fragments of it, possibly with additional large cardinal hypotheses. One direction in this area will be to develop the theory provided by very strong large cardinal hypothesis in the region of a non-trivial elementary embedding from the universe into itself, in a context with little or no Choice, with the attractive goal in mind of deriving a possible contradiction.