Mathematics

We construct a large family of normal κ -complete R κ -embeddable κ + -Aronszajn trees which have no club isomorphic subtrees using an instance of the proxy principle of Brodsky-Rinot.

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce.

For every natural number n we introduce a new weak choice principle nR C fin : Given any infinite set x , there is an infinite subset y?�x and a selection function f that chooses an n -element subset from every finite z?�y containing at least n elements. By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of nR C fin to the weak choice principles R C m (that has already been studied by Montenegro, Halbeisen and Tachtsis): Given any infinite set x , there is an infinite subset y?�x with a choice function f on the family of all m -element subsets of y . Moreover, we prove a stronger analogue of Montenegros results when we study the relation between nR C fin and k C ??fin which is defined by: Given any infinite family F of finite sets of cardinality greater than k , there is an infinite subfamily A?�F with a selection function f that chooses a k -element subset from each A?�A .

Recently there have been numerous proposed solutions to the problem of logical omniscience in doxastic and epistemic logic. Though these solutions display an impressive breadth of subtlety and motivation, the crux of these approaches seems to have a common theme-minor revisions around the ubiquitous Kripke semantics-rooted approach. In addition, the psychological mechanisms at work in and around both belief and knowledge have been left largely untouched. In this paper, we cut straight to the core of the problem of logical omniscience, taking a psychologically-rooted approach, taking as bedrock the "quanta" of given percepts, qualia and cognitions, terming our approach "PQG logic", short for percept, qualia, cognition logic. Building atop these quanta, we reach a novel semantics of belief, knowledge, in addition to a semantics for psychological necessity and possibility. With these notions we are well-equipped to not only address the problem of logical omniscience but to more deeply investigate the psychical-logical nature of belief and knowledge.

We give a p -adic example of a structure whose Shelah completion interprets Q p but which does not (provided an extremely plausible conjecture holds) interpret an infinite field. In the final section we discuss the significance of such examples for a possible future geometric theory of NIP structures.

Mathematicians like Markov and Bishop made an effort to develop constructive mathematics and extended many theorems in classical mathematical analysis. Heine Borel theorem tells us that a closed bounded subset of Euclidean space R is compact, but in constructive mathematics, Tseitin and Zaslavskii showed that the set of all constructive real numbers between 0 and 1 is not compact. We are going to show that when giving certain restriction to the open cover on [0,1], we can however always choose a finite sub-cover.

We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

We prove some unconditional cases of the Existential Closedness problem for the modular j -function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular j -function and its derivatives, one can define a natural closure operator in three equivalent different ways.

We show that Carmo and Jones' condition 5(e) conflicts with the other conditions on their models for contrary-to-duty obligations. We then propose a resolution to the conflict.

We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside this we find "relaxed" set theory, which is much easier to use with full precision than traditional axiomatic theories. There is also an implementation of the Zermillo-Fraenkel-Choice axioms that is maximal in the sense that any other implementation uniquely embeds in it.