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In this thesis we investigate the properties of a group of axioms known as 'Guessing Axioms,' which extend the standard axiomatisation of set theory, ZFC. In particular, we focus on the axioms called 'diamond' and 'club,' and ask to what extent properties of the former hold of the latter. A question of 1. Juhasz, of whether club implies the existence of a Suslin tree, remains unanswered at the time of writing and motivates a large part of our in- vestigation into diamond and club. We give a positive partial answer to Juhasz's question by defining the principle Superclub and proving that it implies the exis- tence of a Suslin tree, and that it is weaker than diamond and stronger than club (though these implications are not necessarily strict). Conversely, we specify some conditions that a forcing would have to meet if it were to be used to provide a negative answer, or partial answer, to Juhasz's question, and prove several results related to this. We also investigate the extent to which club shares the invariance property of diamond: the property of being formally equivalent to many of its natural strength- enings and weakenings. We show that when certain cardinal arithmetic statements hold, we can always find different variations on club t.hat will be provably equiv- alent. Some of these hold in ZFC. But, in the absence of the required cardinal arithmetic, we develop a general method for proving that most variants of club are pairwise inequivalent in ZFC.

Motivated by the papers [84, 85], this thesis considers the concepts of reactivity, Lyapunov stability and Turing patterns. We introduce the notion of P-reactivity, a new measure for transient dynamics. We extend a result by Shorten and Narendra [108] regarding joint dissipativity for second order systems. We derive an easy verifiable formula that determines systems P-reactivity with respect to a norm induced by the positive definite matrix P. An optimization problem aiming to determine the positive definite P with respect to which a stable system is most reactive is posed and solved numerically for second order systems. The stability radius is adopted as a measure of robustness of joint disspaptivity. We characterise the stability radius of joint dissipativity when the underlying systems are subject to certain specific perturbation structures. A detailed robustness analysis of the Shorten and Narendra conditions is also presented. Using the notion of common Lyapunov function we show that the necessary condition in [85] is a special case of a more powerful (i.e tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We also further extend this to include more complicated movement mechanisms such as chemotaxis. Unlike the traditional techniques, this new necessary condition can be used to check Turing instability for systems with any dimension and any number of parameters. We apply our new conditions to various models in literature.

The present thesis is an investigation on an open problem in mathematical logic: the problem of devising an explanation of the meaning of the intuitionistic first-order logical operators, which is both mathematically rigorous and faithful to the interpretation intended by the intuitionistic mathematicians who invented and have been using them. This problem has been outstanding since the early thirties, when it was formulated and addressed for the first time. The thesis includes a historical, expository part, which focuses on the contributions of Kolmogorov, Heyting, Gentzen and Kreisel, and a long and detailed discussion of the various interpretations which have been proposed by these and other authors. Special attention is paid to the decidability of the proof relation and the introduction of Kreisel's extra-clauses, to the various notions of 'canonical proof' and to the attempt to reformulate the semantic definition in terms of proofs from premises. In this thesis I include a conclusive argument to the effect that if one wants to withdraw the extra-clauses then one cannot maintain the concept of 'proof as the basic concept of the definition; instead, I describe an alternative interpretation based on the concept of a construction 'performing' the operations indicated by a given sentence, and I show that it is not equivalent to the verificationist interpretation. I point out a redundancy in the internal-pseudo-inductive-structure of Kreisel's interpretation and I propose a way to resolve it. Finally, I develop the interpretation in terms of proofs from premises and show that a precise formulation of it must also make use of non-inductive clauses, not for the definition of the conditional but -surprisingly enough- for the definitions of disjunction and of the existential quantifier.

Effectively deciding the satisfiability of logical sentences over structures is an area well-studied in the case of finite structures. There has been growing work towards considering this question for infinite structures. In particular the theory of automatic structures, considered here, investigates structures representable by finite automata. The closure properties of finite automata lead naturally to algorithms for deciding satisfiability for some logics. The use of finite automata to investigate infinite structures has been inspired by the interplay between the theory of finite automata and the theory of semigroups. This inspiration has come in particular from the theory of automatic groups and semigroups, which considers (semi)groups with regular sets of normal forms over their generators such that generator-composition is also regular. The work presented here is a contribution to the foundational problem for automatic structures: given a class of structures, classify those members that have an automatic presentation. The classes considered here are various interesting subclasses of the class of finitely generated semigroups, as well as the class of Cayley Graphs of groups. Although similar, the theories of automatic (semi)groups and automatic presentation differ in their construction. A classification for finitely generated groups allows a direct comparison of the theory of automatic presentations with the theory of automatic groups.

In this work I develop a formalization (M-L) of Martin-Löf type theory, the main concern being an accurate definition of what it is to be a model of M-L. Using this definition, I proceed with actual models of M-L (mainly realizability models) to establish the relative consistency of many intuitionistic principles. In addition to their consistency I investigate their interrelationship. The strongest principles which are shown to be consistent with M-L are Church's Thesis and the Fan Theorem. The expressive power of M-L is used to formalize certain theories. The Theory of Real Numbers, an Intuitionistic Set Theory and Category Theory are all formalizable. The constructions of the latter are used to describe Kripke Models of M-L. Finally, I prove that the subsystem of M-L obtained by dropping the rules for cartesian products of types and without rules for universes of types has proof theoretic ordinal ω^ω.

The main topic of this thesis is the generalization to ordered groupoids of some results and constructions that have arisen in groupoid theory and its applications in homological and homotopical algebra. We study fibrations of ordered groupoids, and show that the covering homotopy property and star-surjectivity are not equivalent properties. We establish some formal properties of functors having these properties, and define a new quotient construction for ordered groupoids that leads to a factorization of any functor of ordered groupoids as a star-surjective followed by a star-injective functor. We give a direct proof of Ehresmann’s Maximum Enlargement Theorem. Coupled with our quotient construction, The Maximum Enlargement Theorem gives a universal factorization of any functor of ordered groupoids as a fibration followed by an enlargement followed by a covering. We construct the mapping cocylinder M of an ordered functor : G ! H, and show directly that the morphism M ! H has the covering homotopy property. We construct the derived module D of an ordered functor and use it to study two adjoint functors between the category of ordered crossed complexes and the category of ordered chain complexes. Finally, we consider the groupoid of derivations of crossed modules of groups and of ordered groupoids, and in the latter case we use semiregular crossed modules to derive results on homotopies and endomorphisms. (Mathematical symbols not available - please refer to the PDF).

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