A Sacks-like model with large continuum

In the late 19th century, Georg Cantor proved that the set of all real numbers (such as Pi or the square root of 2) cannot be put into a one-to-one correspondence with the set of all natural numbers (0,1,2,3 etc.). This means that the set of real numbers is, in some sense, "larger" than the set of natural numbers. This breakthrough marked the beginning of set theory, as it showed that infinite sets exhibit interesting and complex mathematical behavior. Building on Cantor`s work, many mathematicians worked to deepen our understanding of the size of the set of real numbers, also called the size of the continuum. A pivotal point was reached when, in the early 1960s, Paul Cohen proved that the Continuum Hypothesis, considered one of the most important open problems in mathematics, is unsolvable. He showed that we cannot compute precisely the size of the continuum using standard mathematical rules. The technique used by Cohen to achieve his result is called forcing. As the development of this technique significantly advanced during the 1960s and 1970s, it led to new and striking unsolvability results but also revealed some challenging phenomena. One of such issues is the seemingly structural difficulty in showing that a vast class of mathematical statements (like "Every big enough set of reals can be mapped continuously onto the unit interval [0,1]") do not imply the existence of a bound on the size of the continuum. Many of the most important open questions in modern set theory revolve around overcoming this problem. Our project aims to contribute to this research by developing a new framework for iterating a particular kind of forcing known as Jensen`s forcing, introduced by Ronald B. Jensen in 1970. This work has the potential to significantly enhance our understanding of the structural properties of the continuum.