Mathematics

The goal of this introductory survey is to present the major developments of algorithmic randomness with an eye toward its historical development. While two highly comprehensive books and one thorough survey article have been written on the subject, our goal is to provide an introduction to algorithmic randomness that will be both useful for newcomers who want to develop a sense of the field quickly and interesting for researchers already in the field who would like to see these results presented in chronological order.

The methodology used here might provide a neat method of examining paradoxes and ways to circumvent them. Most of the known set theoretic paradoxes (Russell's, Cantor's, Burali-Forti's,..) can be paralleled here and examined. This account will shed the light on a particular application of this method that appears to elude paradoxes; an application that have shortcomings that will be illustrated here and suggestions to solutions are proposed. First I'll present the exposition of the theory, and then speak about its background and the aims behind it and how it can extend our knowledge of overcoming paradoxes.

We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exissts a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.

Building on work of Holy, Lücke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of several well-known large cardinal ideals, including the Ramsey ideal, in terms of generic elementary embeddings; we also point out some seemingly inherent differences between small embedding and generic embedding characterizations of subtle cardinals. Additionally, we present a simple and uniform proof which shows that, when κ is weakly compact, many large cardinal ideals on κ are nowhere κ -saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and ? 1 1 -indescribable ideals on P κ λ , and we close with a list of open questions.

For every univariate formula χ we introduce a lattices of intermediate theories: the lattice of χ -logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula χ 2 , which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of χ -logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.

We prove that any suitable generalization of Laver forcing to the space κ κ , for uncountable regular κ , necessarily adds a Cohen κ -real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if κ <κ =κ , then every <κ -distributive tree forcing on κ κ adding a dominating κ -real which is the image of the generic under a continuous function in the ground model, adds a Cohen κ -real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140

This paper aims at discussing the importance of Leibniz Law to getting models for Paraconsistent Set Theories.

We introduce a principle of local collection for compositional truth predicates and show that it is conservative over the classically compositional theory of truth in the arithmetical setting. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the case of induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate.

We prove that every locally constant constructive function on an interval is in fact a constant function. This answers a question formulated by Andrej Bauer. As a related result we show that an interval consisting of constructive real numbers is in fact connected, but can be decomposed into the disjoint union of two sequentially closed nonempy sets.

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.