Boolean-valued structures

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

Standardization principle of nonstandard universes

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.

An elementary definability theorem for first order logic

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.

Sheaves and Boolean valued model theory

In recent years model theorists have been studying various sheaf-theoretic notions as they apply to model theory. For quite a while however, a sheaf of structures was considered to be just a local homeomorphism between topological spaces such that each stalk Sx = p−1(x) is a model-theoretic structure and such that certain maps are continuous. Some of the model-theoretic work done with this notion of a sheaf of structures are the papers by Carson [2] and Macintyre [7]. Soon came the idea of considering a sheaf of structures not just as a collection of structures glued together in some continuous way, but rather as some sort of generalized structure. A significant model-theoretic study of sheaves in this new sense became possible only after the development of the theory of topoi. As F.W. Lawvere pointed out in [6], this represents the advance of mathematics (in our case the advance of model theory) from metaphysics to dialectics.A topos is the rather ingenious evolution of the notion of a Grothendieck topos [13]. It provides us with the idea that an object of a topos (e.g. the topos of sheaves over a topological space) may be thought of as a generalized set. Furthermore, all first-order logical operations have an interpretation in a topos, hence we may talk about generalized structures. Angus Macintyre suggested that some of his model-theoretic results about sheaves of structures may be understood better and perhaps simplified by doing model theory inside a topos of sheaves.

Sheaves and normal submodels

Ellerman, Comer, and Macintyre have all observed that sheaves are an interesting generalization of models and are deserving of model theoretic attention. Scott has pointed out that sheaves are Heyting algebra valued models. The reverse does not hold however since almost no genuine Boolean valued model is a sheaf.In §1 we shall review the definition of a sheaf and prove a theorem about Boolean valued models using the sheaf construction. In §2 we shall be concerned with the set of sentences preserved by global sections. Our principal result is that global section sentences are also normal submodel sentences. (We define as a normal submodel of if is a submodel of and every point of B − A can be moved by an automorphism of which fixes each point of A.) In §3 we prove that every normal submodel sentence is the negation of a disjunction of Horn sentences and that the set of normal submodel sentences is r.e. but not recursive. §3 involves only traditional model theory and can be read independently of the first two sections.