Mathematics

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) {\em Boolean algebra of conditionals} from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.

Boolean valued models for a signature L are generalizations of L -structures in which we allow the L -relation symbols to be interpreted by boolean truth values; for example for elements a,b∈M with M a B -valued L -structure for some boolean algebra B , (a=b) may be neither true nor false, but get an intermediate truth value in B . In this paper we expand and relate the work of Mansfield and others on the semantics of boolean valued models, and of Munro and others on the adjunctions between B -valued models and B + -presheaves for a boolean algebra B . In particular we give an exact topological characterization (the so called \emph{fullness property}) of which boolean valued models satisfy Łoś theorem (i.e. the general form of the forcing theorem which Cohen -- Scott, Solovay, Vopenka -- established for the special case given by the forcing method in set theory). We also give an exact categorial characterization of which presheaves correspond to \emph{full} boolean valued models in terms of the structure of global sections of their associated étalé space. To do so we introduce a slight variant of the sheafification process of a given presheaf by means of its embedding into an étalé space.

This article explores the connection between boolean-valued class models of set theory and the theory of arbitrary objects in roughly Kit Fine's sense of the word. In particular, it explores the hypothesis that the set theoretic universe as a whole can be seen as an arbitrary entity, which can in turn be taken to consist of arbitrary objects, viz. arbitrary sets.

Every better quasi-order codifies a Borel graph that does not contain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic number, though one has yet to be explicitly constructed. In this paper, we show that examples cannot be constructed via standard methods. Moreover, we show that most of the known better quasi-orders are non-examples, suggesting there is still a class of better quasi-orders with interesting combinatorial properties who's elements/members still remain unknown.

We show that Vizing's Theorem holds in the Borel context for graphs induced by actions of 2-ended groups, and ask whether it holds more generally for everywhere two ended Borel graphs.

We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups (H,+) . In particular, we show that if α< ω 1 and 4≤k<ω , then there is a ccc forcing notion adding a Σ 0 2 set B⊆H which has ℵ α many pairwise k --overlapping translations but not a perfect set of such translations.

Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author. Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bounded symbiosis. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem-type principles for second order logic.

This paper introduces Broad Infinity, a new and arguably intuitive axiom scheme. It states that "broad numbers", which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal. Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The paper relates these principles under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.

It is investigated in what sense the Brouwer fixed point theorem may be viewed as a corollary of the Lawvere fixed point theorem. A suitable generalisation of the Lawvere fixed point theorem is found and a means is identified by which the Brouwer fixed point theorem can be shown to be a corollary, once an appropriate continuous surjective mapping A ′ → X A ′′ has been constructed for each space X in a certain class of "nice" spaces for each one of which the exponential topology on X A ′′ exists, and here A ′ and A ′′ have the same carrier set and the topology on A ′ is finer than on A ′′ . It is shown that there is a certain natural way of attempting to derive Brouwer as a corollary of Lawvere which is not possible, that is there is no space A for which the exponential topology on [0,1 ] A exists and there is a continuous surjection A→[0,1 ] A . We then examine the range of contexts in which phenomena like those described in the first result occur, from a broadly model-theoretic perspective, with a view towards applications for the original motivation for the problem as a problem in decision theory for AI systems, suggested by the Machine Intelligence Research Institute.