# Tutorial on Boolean ultrapowers, BLAST 2015, Las Cruces, NM

I shall give a tutorial lecture series on Boolean ultrapowers, two or three lectures, at the BLAST conference in Las Cruces, New Mexico, January 5-9, 2015.  (The big AMS meeting in San Antonio, reportedly a quick flight, begins on the 10th.)

BLAST is a conference series focusing on

B = Boolean Algebras
L = Lattices, Algebraic and Quantum Logic
A = Universal Algebra
S = Set Theory
T = Set-theoretic and Point-free Topology

In this tutorial, I shall give a general introduction to the Boolean ultrapower construction.

Boolean ultrapowers generalize the classical ultrapower construction on a power-set algebra to the context of an ultrafilter on an arbitrary complete Boolean algebra. Introduced by Vopěnka as a means of undertaking forcing constructions internally to ZFC, the method has many connections with forcing. Nevertheless, the Boolean ultrapower construction stands on its own as a general model-theoretic construction technique, and historically, researchers have come to the Boolean ultrapower concept from both set theory and model theory.  An emerging interest in Boolean ultrapowers arises from a focus on well-founded Boolean ultrapowers as large cardinal embeddings.

In this tutorial, we shall see that the Boolean ultrapower construction reveals that two central set-theoretic techniques–forcing and classical ultrapowers–are facets of a single underlying construction, namely, the Boolean ultrapower.  I shall provide a thorough introduction to the Boolean ultrapower construction, assuming only an elementary graduate student-level familiarity with set theory and the classical ultrapower and forcing techniques.

# Inner models with large cardinal features usually obtained by forcing

• A. Apter, V. Gitman, and J. D. Hamkins, “Inner models with large cardinal features usually obtained by forcing,” Archive for Mathematical Logic, vol. 51, pp. 257-283, 2012. (10.1007/s00153-011-0264-5)
@article {ApterGitmanHamkins2012:InnerModelsWithLargeCardinals,
author = {Apter, Arthur and Gitman, Victoria and Hamkins, Joel David},
affiliation = {Mathematics, The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY 10016, USA},
title = {Inner models with large cardinal features usually obtained by forcing},
journal = {Archive for Mathematical Logic},
publisher = {Springer Berlin / Heidelberg},
issn = {0933-5846},
keyword = {Mathematics and Statistics},
pages = {257--283},
volume = {51},
issue = {3},
url = {http://jdh.hamkins.org/innermodelswithlargecardinals/},
eprint = {1111.0856},
archivePrefix = {arXiv},
primaryClass = {math.LO},
doi = {10.1007/s00153-011-0264-5},
note = {10.1007/s00153-011-0264-5},
year = {2012}
}

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal $\kappa$ for which $2^\kappa=\kappa^+$, another for which $2^\kappa=\kappa^{++}$ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model $W$ with a strongly compact cardinal $\kappa$, such that $H_{\kappa^+}^V\subseteq HOD^W$. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit $\delta$ of ${\lt}\delta$-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

# An introduction to Boolean ultrapowers, Bonn, 2011

A four-lecture tutorial on the topic of Boolean ultrapowers at the Young Set Theory Workshop at the Hausdorff Center for Mathematics in Königswinter near Bonn, Germany,  March 21-25, 2011.

Boolean ultrapowers generalize the classical ultrapower construction on a power-set algebra to the context of an ultrafilter on an arbitrary complete Boolean algebra. Closely related to forcing and particularly to the use of Boolean-valued models in forcing, Boolean ultrapowers were introduced by Vopenka in order to carry out forcing as an internal ZFC construction, rather than as a meta-theoretic argument as in Cohen’s approach. An emerging interest in Boolean ultrapowers has arisen from a focus on the well-founded Boolean ultrapowers as large cardinal embeddings.

Historically, researchers have come to the Boolean ultrapower concept from two directions, from set theory and from model theory. Exemplifying the set-theoretic perspective, Bell’s classic (1985) exposition emphasizes the Boolean-valued model $V^{\mathbb{B}}$ and its quotients $V^{\mathbb{B}}/U$, rather than the Boolean ultrapower $V_U$ itself, which is not considered there. Mansfield (1970), in contrast, gives a purely algebraic, forcing-free account of the Boolean ultrapower, emphasizing its potential as a model-theoretic technique, while lacking the accompanying generic objects.

The unifying view I will explore in this tutorial is that the well-founded Boolean ultrapowers reveal the two central concerns of set-theoretic research–forcing and large cardinals–to be two aspects of a single underlying construction, the Boolean ultrapower, whose consequent close connections might be more fruitfully explored. I will provide a thorough introduction to the Boolean ultrapower construction, while assuming only an elementary graduate student-level familiarity with set theory and the classical ultrapower and forcing techniques.