Translation Procedures and Derived Models

Zermelo-Fraenkel Set Theory forms the foundation for modern mathematics. By Gödels famous Incompleteness Theorem there are statements that cannot be settled in this system. Since then set theorists have studied a wide variety of new axioms that might one day allow us to find answers to many questions left open by the currently accepted axioms. The strength of a particular axiom can be measured against a specific class of axioms that posit the existence of particularly large infinities known simply as large cardinals. There is another class of axioms with similar properties: axioms of determinacy. These posit that certain two- player perfect information games played on natural numbers are determined, i.e. one of the two players has a strategy which guarantees their victory in any play of the game. One set of axioms concerns itself with the largest possible structures while the other talks about objects which are by the standards of infinities quite small. Yet, there exists a deep equivalence between them which is unlocked through the study of canonical models of set theory. This topic can be traced back to Kurt Gödels work on the consistency of the continuum hypothesis, but was only truly unlocked through the fine structure analysis of Ronald B. Jensen. Jensen also coined the term we still use for these structures: mice. One important facet of this equivalence is W.H. Woodin`s Derived Model Theorem through which we can assign models of determinacy to mice. Our project aims to uncover some of the key mechanics by which the structure of the given mouse determines the structure of an associated derived model. This will depend heavily on creating procedures that translate between `classical mice and HOD mice, a variant developed by Sargsyan, Steel, Woodin et al which appears naturally in the study of models of determinacy. Progress here will furthermore allow us to tackle many crucial problems at the current frontiers of the study of infinity.