Boolean ultrapowers

Bibliography: leaves 121-122.The Boolean ultrapower construction is a generalisation of the ordinary ultrapower construction in that an arbitrary complete Boolean algebra replaces the customary powerset Boolean algebra. B. Koppelberg and S. Koppelberg [1976] show that the class of ordinary ultrapowers is properly contained in the class of Boolean ultrapowers thereby justifying the development of a theory for Boolean ultrapowers. This thesis is an exploration into the strategies whereby and the conditions under which aspects of the theory of ordinary ultrapowers can be extended to the theory of Boolean ultrapowers. Mansfield [1971] shows that a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower under certain conditions. Using a different approach and under somewhat different conditions, Ouwehand and Rose [1998] show that the result also holds for K-bounded Boolean ultrapowers. Mansfield [1971] also proves a Boolean version of the Keisler-Shelah theorem. By redefining the notion of a K-good ultrafilter on a Boolean algebra, Benda [1974] obtains a complete generalisation of a theorem of Keisler which states that an ultrapower is K-saturated iff the ultrafilter is K-good. Potthoff [1974] defines the notion of a limit Boolean ultrapower and shows that, as is the case for ordinary ultrapowers, the complete extensions of a model are characterised by its limit Boolean ultrapowers. Upon the discovery by Frayne, Morel and Scott [1962] of an ultrapower of a simple group which is not simple, Burris and Jeffers [1978] investigate necessary and sufficient conditions for a Boolean ultrapower to be simple, or subdirectly irreducible, provided that the language is countable. Finally, Jipsen, Pinus and Rose [2000] extend the notion of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, and prove that by using this definition, Blass' Characterisation Theorem can be generalised for Boolean ultrapowers

Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

From quantum to cosmological regime. The role of forcing and exotic 4-smoothness

Recently, a cosmological model based on smooth open 4-manifolds admitting non-standard smoothness structures was proposed. The manifolds are exotic versions of R4 and S3 R. The model has been developed further and proven to be capable of obtaining some realistic cosmological parameters from these exotic smoothings. The important problem of the quantum origins of the exotic smoothness of space-time is addressed here. It is shown that the algebraic structure of the quantum-mechanical lattice of projections enforces exotic smoothness on Rn. Since the only possibility for such a structure is exotic R4, it is found to be a reasonable explanation of the large-scale four-dimensionality of space-time. This is based on our recent research indicating the role of set-theoretic forcing in quantum mechanics. In particular, it is shown that a distributive lattice of projections implies the standard smooth structure on R4. Two examples of models valid for cosmology are discussed. The important result that the cosmological constant can be identified with the constant curvature of the embedding (exotic R4) ! R4 is referred. . The calculations are in good agreement with the observed small value of the dark energy density

Some Applications of Set Theory to Model Theory

We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity. Keisler's order is a pre-order on complete countable theories $T$, measuring the saturation of ultrapowers of models of $T$. In Chapter~\ref{SurveyChapter}, we present a self-contained survey on Keisler's order. In Chapter~\ref{KeislerNew}, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's order among the simple unstable theories. Borel complexity is a pre-order on sentences of $\mathcal{L}_{\omega_1 \omega}$ measuring the complexity of countable models. In Chapter~\ref{ChapterURL}, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of $\Phi \in \mathcal{L}_{\omega_1 \omega}$ with the number of potential canonical Scott sentences of $\Phi$. In Chapter~\ref{ChapterSB}, we introduce the notion of thickness; when $\Phi$ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter~\ref{ChapterTFAG}, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups