Mathematics

We exhibit how the Rasiowa-Sikorski Lemma simplifies, in a sense, proofs of results that make use of the technique known as back-and-forth, often resulting in not very illustrative arguments. The first two sections seek to show one simple and one complicated proofs of known results, in the hopes that the reader appreciates how the arguments end up, in our view, considerably clearer than those found in classic literature. The final section shows how the same techniques can be adapted to areas commonly considered distant to Set Theory, in this instance, Graph Theory. Sections one and two are based on my bachelor thesis, under the direction of Dr. Roberto Pichardo Mendoza, whom I deeply thank for his advice and revision of this work. All results mentioned in this paper are well known, however, as far as we know, the proofs in the last two sections are original.

Inspired by Ramsey's theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph G contains an infinite subset H such that every vertex of G is adjacent to precisely none, one, or infinitely many vertices of H . We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramsey's theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramsey's theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorem's computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak König's lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramsey's theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramsey's theorem for pairs by showing that a number of well-known consequences of Ramsey's theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem.

Interpretability logics are endowed with relational semantics à la Kripke: Veltman semantics. For certain applications though, this semantics is not fine-grained enough. Back in 1992, in the research group of de Jongh, the notion of generalised Veltman semantics emerged to obtain certain non-derivability results as was first presented by Verbrugge ([76]). It has turned out that this semantics has various good properties. In particular, in many cases completeness proofs become simpler and the richer semantics will allow for filtration arguments as opposed to regular Veltman semantics. This paper aims to give an overview of results and applications of Generalised Veltman semantics up to the current date.

We consider a 2-valued non-deterministic connective ?�∨ defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it platypus. The value of ?�∨ is not completely determined by the input, contrasting with usual notion of Boolean connective. We call non-deterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. Unexpectedly, this very simple connective and the logic it defines, illustrate various key advantages in working with generalized notions of semantics (by incorporating non-determinism), calculi (by allowing multiple-conclusion rules) and even of logic (moving from Tarskian to Scottian consequence relations). We show that the associated logic cannot be characterized by any finite set of finite matrices, whereas with non-determinism two values suffice. Furthermore, this logic is not finitely axiomatizable using single-conclusion rules, however we provide a very simple analytical multiple-conclusion axiomatization using only two rules. Finally, deciding the associated multiple-conclusion logic is coNP-complete, but deciding its single-conclusion fragment is in P.

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders {o(μ)???λ} is unbounded in λ . Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

The Bristol model is an inner model of L[c] , where c is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This paper is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of ZF . We also correct a few minor issues in the original paper, as well as prove new results. For example, that the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and that the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna--Wagner Principles, which we think may play an important role in understanding the generic multiverse in ZF .

To strengthen the effectiveness of approximate reasoning in fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) problems, three approximate reasoning schemes with aggregation functions are developed and their validity is respectively investigated in this paper. We firstly study some properties of fuzzy implication generated by aggregation function. And then present an A -compositional rule of inference (ACRI) as an extension of Zadeh's CRI replacing t -norm by an aggregation function. The similarity-based approximate reasoning with aggregation functions is further discussed. Moreover, we provide the quintuple implication principle (QIP) method with aggregation functions to solve FMP and FMT problems. Finally, the validity of our proposed three approximate reasoning approaches is respectively analyzed using GMP rules in detail.

The notion of a critical successor [dJV90] has been central to almost all modal completeness proofs in interpretability logics. In this paper we shall work with an alternative notion, that of an assuring successor. As we shall see, this will enable more concisely formulated completeness proofs, both with respect to ordinary and generalised Veltman semantics. Due to their interesting theoretical properties, we will devote some space to the study of a particular kind of assuring labels, the so-called full labels. After a general treatment of assuringness, we shall apply it to obtain certain completeness results. Namely, we give another proof of completeness of ILW w.r.t. ordinary semantics and of ILP w.r.t. generalised semantics.

The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer d and a subset A⊆Z whose set of representations base d is recognized by a finite automaton, a necessary condition is found for x+y∈A to be a stable formula in Th(Z,+,A) . Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the d -automatic A⊆Z such that (Z,+,A) is stable. This characterization is in terms of what were called " F -sets" by Moosa and Scanlon and "elementary p -nested sets" by Derksen. Automata-theoretic methods are also used to produce some NIP expansions of (Z,+) , in particular the expansion by the monoid ( d N ,×) .

We describe two systems for supporting beginner students in acquiring basic skills in expressing statements in the formalism of first-order predicate logic; the first, called "math dictations", presents users with the task of formalizing a given natural-language sentence, while the second, called "Game of Def", challenges users to give a formal description of a set of a geometric pattern displayed to them. In both cases, an automatic checking takes place.