# Blog Archives

# Topic Archive: Inverse limits

# Reflecting I_0

We will present an argument for reflecting the large cardinal axiom I_0 from marginally stronger large cardinals. This will involve presenting some of the theory of inverse limits, which R. Laver first studied in the context of reflecting large cardinals at this level. Along the way we will see many local reflection results below I_0 and state a strong form of reflection which is useful in other contexts.

# Dissertation Defense: Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal

This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved in both directions by forcing under fairly general assumptions. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding $j: V to M$ such that $M$ is closed under sequences of length $supset{j(f)(kappa) st f: kappa to kappa}$. Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.