Long extenders, Varsovian models, Combinatorics

• Applicant: Professor Farmer Schlutzenberg, Ph.D.
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 445387776

Set theory provides a rigorous foundation for most of modern mathematics. The standard axioms (denoted ZFC) are, however, incomplete, and much research in set theory is focused on understanding and comparing various strengthenings of ZFC. These strengthenings come in various forms. A key central form consists in the large cardinal axioms, which posit the existence of strong forms of infinity and reflection. There are intricate connections between such axioms and other seemingly unrelated axioms, which deal with topics such as winning strategies in infinite games and direct combinatorial principles.Inner model theory provides some of our deepest understanding of large cardinal axioms. Given a large cardinal axiom A, one would like to find a set-theoretic universe M_A in which A is true and M_A is the minimal natural instance of this axiom. We call such models M_A mice. Mice M are built from a sequence E^M of extenders E, which are highly complex sets encoding large cardinal information. If we allow a biological analogy, the extender sequence can be thought of as like the DNA of a mouse.Inner model theorists have constructed mice with Woodin cardinals, a mid-strength large cardinal. The key obstacle to constructing more complex mice is that we lack a proof that those built are iterable. Iterability implies, roughly, that the extender sequence E^M can be stretched out and examined carefully, and thereby compared with the extender sequences of other mice. Iterability requires the existence of an iteration strategy (which succeeds in the stretching process) and involves dealing with (potentially highly complex) iteration trees. Iterability is used in many ways.Ignoring the actual construction of mice, one can simply posit that a mouse satisfying a given large cardinal axiom A exists, and study its properties. There is a rich theory of such analysis. The goal of the project is to solve problems within (some of) 3 parts of inner model theory. These are:1. Extending particular methods for short extender mice to those built from long extenders. (Long extenders are more complex than their short counterparts. Various methods have been developed for short extenders only, and it would be very interesting to extend them to the long.)2. Addressing specific questions associated with Varsovian models. (This is a recently discovered kind of structure which can appear inside a mouse, and which the mouse can use to compute and understand much of its own iteration strategy, more than might have been previously expected. Varsovian models also constitute somehow the most canonical "core" of a mouse.)3. Considering certain combinatorial questions and their connection to inner models. (One goal here is to study certain diamond-principle combinatorics inside mice - diamonds can ``predict'' well the contents of many other sets. Another is to consider the influence of mice on combinatorics in the more general universe.)

• DFG Programme: Research Grants
• International Connection: Austria