Online research in philosophy

There is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some model-theoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.

Minimal entities are, roughly, those that fall under notions defined by only deflationary principles. In this paper I provide an accurate characterization of two types of minimal entities: minimal properties and minimal facts. This characterization is inspired by both Schiffer's notion of a pleonastic entity and Horwich's notion of minimal truth. I argue that we are committed to the existence of minimal properties and minimal facts according to a deflationary notion of existence, and that the appeal to the inferential role reading of the quantifiers does not dismiss this commitment. I also argue that deflationary existence is language-dependent existence—this clarifies why minimalists about properties and facts are not realists about these entities though their language may appear indistinguishable from the language of realists.

All the major inter-theoretic relations of fundamental science are asymptotic ones, e.g. quantum theory as Planck's constant h 0, yielding (roughly) Newtonian mechanics. Thus asymptotics ultimately grounds claims about inter-theoretic explanation, reduction and emergence. This paper examines four recent, central claims by Batterman concerning asymptotics and reduction. While these claims are criticised, the discussion is used to develop an enriched, dynamically-based account of reduction and emergence, to show its capacity to illuminate the complex variety of inter-theory relationships in physics, and to provide a principled resolution to such persistent philosophical problems as multiple realisability and the nature of the special sciences. Introduction Exposition Examination I: Claims (1) and (2), asymptotic explanation and reference Examination II: Claim (3), reduction and singular asymptotics Examination III: Claim (4), emergence and multiple realisability Conclusion.

For a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model M of T is sequentially complete if and only if $\mathscr{M} \prec \mathscr{N}$ for some Dedekind complete model N. We also prove that if T has a Dedekind complete model of power greater than 2 ℵ 0 , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory--namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables--that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

The existence of a countable complete theory with exactly ℵ 1 many minimal models is independent of ZFC + ¬CH.

For any countable transitive complete theory T with infinite models and the finite model property, we construct a minimal structure M such that the theory of M is small if and only if T is small, and is λ-stable if and only if T is λ-stable. This gives a series of new examples of minimal structures.

The notion of quasi-minimal structures was defined by B. Zil'ber as a natural generalization of minimal structures. Inspired by his work, we study here basic model theoretic properties of quasiminimal structures. Main result is the construction of ω-saturated quasi-minimal models under ω-stability assumption.

Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it.

Answering a problem of Fuhrken we prove that for every κ, 1 ≤ κ ≤ ℵ 0 , there is a (countable) complete theory T, with no prime model, and exactly κ minimal models (up to isomorphism).

It is argued that one can learn from minimal economic models. Minimal models are models that are not similar to the real world, do not resemble some of its features, and do not adhere to accepted regularities. One learns from a model if constructing and analysing the model affects one’s confidence in hypotheses about the world. Economic models, I argue, are often assessed for their credibility. If a model is judged credible, it is considered to be a relevant possibility. Considering such relevant possibilities may affect one’s confidence in necessity or impossibility hypotheses. Thus, one can learn from minimal economic models.