Reflection and Compactness in Set Theory

The idea of infinity had fascinated philosophers and mathematicians since the dawn of history. In the 19th century George Cantor started to investigate mathematically the idea of different types of infinite sizes. His ideas led to the development of Set Theory the mathematical theory of collections. The objects which are studied in Set Theory are extremely simple they are just collections (sets) of other collections. Although those objects might appear to be simple, they can be used to construct extremely complicated structures, which are rich enough in order to code (in some sense) almost every field in mathematics. While investigating the different types of infinity, Cantor asked the following question: is there an infinite size strictly between the size of the set of all real numbers and the size of the set of all natural numbers? This question, which is called the Continuum Hypothesis, appeared at the beginning of Hilberts famous list of 23 most important mathematical questions for the 20-th century. Eventually, the continuum hypothesis was shown to be independent of the standard axioms of set theory by Kurt Godel and Paul Cohen. Following this discovery, the focal point of the study of set theory was shifted from asking what is true to asking what is possible or consistent. In this project we are interested in the limits and possibilities of the compactness phenomena. In general, a compactness phenomenon is a situation in which a trait of a large structure appears (reflects) already in some small sub-structure. The possibility of various compactness phenomena in large infinite structures coheres with the philosophical idea that even though there are different sizes of infinity, they are quite similar to each other. On the other hand, limitations on compactness phenomena might help us to obtain a deep understanding of the differences between different sizes of infinity. Compactness phenomena are tightly related to large cardinal axioms. Those axioms surpass the standard axioms of set theory in the sense that they can prove the consistency of assertions which set theory cannot prove. Large cardinal axioms form a hierarchy. One large cardinal axiom is stronger than the other if it can prove that more assertions are consistent. This hierarchy is extremely efficient in measuring exactly the strength of many combinatorial assertions (this type of strength is called the consistency strength). 1 Most of the natural compactness principles have nontrivial consistency strength and a lot of them are expected to have the same consistency strength as some well known large cardinal axioms. One of the goals of this project is to get a deeper understanding of the exact connection between large cardinal axioms and compactness phenomena. 2

In the area of set theory, in mathematical logic, the study of large cardinals and elementary embeddings is fundamental. This project focuses on the notion of strongly compact cardinals and some related issues. Those large cardinals are mysterious in the sense that while having very high consistency strength, their effect on the universe of sets is subtle, and difficult to study. In this project, I studied a few notions related to the connection between inner model theory, covering arguments and elementary embeddings in the presence of large cardinals. This led to some deep questions, related to Woodin`s work on the ultimate goal of inner model theory - the HOD conjecture and the Ultimate-L. This led to a variety of work, with many collaborators. For example, together with Ben-Neria (HUJI), we obtained the consistency of the existence of a model that challenges the HOD conjecture, even though it is still very far from refuting it. These results might illuminate the road to a proof of the HOD- conjecture. In another work, joint with Sandra Müller (KGRC), we obtained a combinatorial property which can be used as a gauge for the ability of well behaved inner models to cover the universe.Another line of research is related to the classical definition of strong compactness using the filter completion property. Following a pair of questions of Gitik (TAU), I studied the problem of a restricted filter extension property. Using purely combinatorial arguments, I obtained a surprising equivalence between a seemingly weak version of the filter completion property, and a much stronger one. Similarly, in a joint work with Magidor (HUJI), we obtained a formulation of near-supercompactness using the existence of branches in certain trees. Both of those results suggest that there is a natural normalization process, moving from the wild strongly compact type behavior to the more well behaving supercompact one.