# Elementary Epimorphisms Talk

Elementary epimorphisms between models of set theory
Norman Lewis Perlmutter, LaGuardia Community College
October 17, 2014
This talk concerns joint work with Robert S. Lubarsky.
Elementary epimorphisms were introduced by Philipp Rothmaler. A surjective homomorphism f: M –> N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters.
Philipp asked me whether nontrivial elementary epimorphisms between models of set theory exist. We answer this question in the negative for fully elementary epimorphisms between models of ZFC, but in the positive under weaker assumptions.
In particular, we show that every Pi_1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Sigma_1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^-.

# Models of set theory

I gave a talk for the Florida Atlantic University math colloquium last Friday, October 25. The abstract is below. Markus Schmidmeier, the colloquium organizer, told me today that a record number of his students attended this talk as compared to other colloquium talks, and that they told him they enjoyed the talk. It was a fun talk to give, very high-level with no proofs, an overview of set theory to encourage students to learn about it more deeply later. I touched on the ZFC axioms, forcing, inner models, and large cardinals.

Abstract: The ZFC axioms are the basic axioms underlying almost work in contemporary mathematics. Set theorists refer to the universe of all sets as V. The sets of V, as a whole, satisfy the ZFC axioms. But is this the only universe of sets? Are there other universes of sets that also satisfy the ZFC axioms? In fact, there are. I will discuss these universes of sets and how set theorists build them. I will also discuss how a statement can be true in one universe of sets but false in another. In this case, we say that the statement is independent of ZFC.

# Dissertation defense, April 12

Today, I defended my dissertation. You can view the slides from the talk here.

Abstract

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 characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given by either the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that 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 by forcing in both directions under fairly general assumptions but not in all cases. 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 –> M such that M is closed under sequences of length equal to the clearance of the embedding. This clearance is defined as the supremum, over all functions f from kappa to kappa, of j(f)(kappa). Two of the most important results in the chapter are as follows. A Vopěnka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.

(This post has been backdated.)

# Equivalence of Vopenka cardinals and Woodinized supercompact cardinals

Set Theory Seminar
I present a tentative result that Woodinized supercompact cardinals (also known as Woodin for supercompactness cardinals) are equivalent to Vopenka cardinals. This result is vaguely hinted at, though not proven, in Kanamori’s text, and I believe I have worked out the details. A cardinal $\kappa$ is Vopenka iff for every collection of kappa many model-theoretic structures with domains elements of $V_\kappa$, there exists an elementary embedding between two of them. A cardinal $\kappa$ is Woodinized supercompact if it meets the definition of a Woodin cardinal, with strongness replaced by supercompactness. That is to say, for every function $f:\kappa \to \kappa$, there exists a closure point $\delta$ of $f$ and an elementary embedding $j:V \to M$ such that $j(\delta)<\kappa$ and $M$ is closed in $V$ under $j(f)(\delta)$ sequences.