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 Vopenka 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.

I have a question regarding Boolean -valued models and Ultrapowers. In 1979

(Journal of Philosophical Logic 8 (1979), pp 509-534), Richard B. White proved the consistency of the unrestricted Axiom of Comprehension in the infinite-valued predicate logic of Lukasiewicz. Can a Boolean-valued model be defined as a type of infinite-valued predicate logic and if so, then can the unrestricted Axiom of Comprehension

be proved consistent in an Boolean-Valued Model (of perhaps better put, can the predicate logic of Lukasiewicz be interpreted as a type of Boolean-valued model)? If so, then what does this mean for Boolean ultrapowers and forcing?

The Boolean-valued models I am discussing here, which do involve multi-valued truth, nevertheless always satisfy ZFC with Boolean value one, and ZFC refutes the general comprehension axiom.

If it is so that Boolean-valued models always satisfy ZFC with Boolean value one, is it then not possible to have Boolean-valued models of, say ZFC without Power Set or Foundation; or of Boolean-valued models which don’t satisfy Choice but satisfy various forms of Determinacy ? If this is the case can the notion of Boolean-valued model be so tweaked so that, say, Foundation is false? What would such Boolean-valued models look like?

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