Combinatorics on the reals and forcing theory

In the 19th century, Georg Cantor proved that there is no bijection from the set of natural numbers onto the power set of natural numbers, called the set of the reals or the continuum, and he conjectured that the size of the set of the reals is equal to the smallest uncountable cardinal. This conjecture is called the Continuum Hypothesis, CH. Many people had tried to solve CH, but in the 20th century, after Zermelo--Fraenkel set theory with the axiom of choice, ZFC, which is considered as one of the mainstream axiomatizations of mathematics, was introduced, Kurt Goedel proved that we cannot prove the negation of CH from ZFC by introducing the constructible universe, and Paul Cohen proved that we cannot prove CH from ZFC by creating the forcing method which is a technique for extending a model of set theory. The forcing method has been developed by a number of set theorists. This development has provided us with many consistency statements not only in set theory but also in analysis, algebra, geometry and other fields of mathematics. The development of forcing theory contributed to our understanding of the set of the reals from the combinatorial point of view. Combinatorics on the reals has been studied before axiomatic set theory has appeared. After ZFC has been introduced and the results due to Goedel and Cohen have been obtained, many statements have been proved to be independent from ZFC using forcing theory. Since the middle of the 20th century, research has been done on cardinal invariants of the reals and the structure of the power set of natural numbers modulo finite to advance our knowledge of the reals. In the early 1990s, W. Hugh Woodin has introduced the forcing notion, called Pmax. He and other people studying with him in the 1990s have developed the Pmax-style forcing notions, called Pmax variations. The goal of the project is to find other inequalities between cardinal invariants of the reals and other properties of the structure of the power set of the natural numbers modulo finite which are provable in ZFC or are independent from ZFC, and to develop the forcing theory needed to show consistency results about cardinal invariants using the approach via Pmax variations.