Thesis (doctoral)--Universität Bonn, 1988. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin, 1964. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Bibliography: leaves 124-128.

Thesis (Ph. D.)--Illinois State University, 1997. / Title from title page screen, viewed June 1, 2006. Dissertation Committee: Albert D. Otto, Cheryl A. Lubinski (co-chairs), John A. Dossey, Cynthia W. Langrall, George Padavil. Includes bibliographical references (leaves 119-123) and abstract. Also available in print.

In this thesis I investigate the implications, for one's account of mathematics, of holding an anti-realist view. The primary aim is to appraise the scope of revision imposed by anti-realism on classical inferential practice in mathematics. That appraisal has consequences both for our understanding of the nature of mathematics and for our attitude towards anti-realism itself. If an anti-realist position seems inevitably to be absurdly revisionary then we have grounds for suspecting the coherence of arguments canvassed in favour of anti-realism. I attempt to defend the anti-realist position by arguing, i) that it is not internally incoherent for anti-realism to be a potentially revisionary position, and ii) that an anti-realist position can, plausibly, be seen to result in a stable intuitionistic position with regard to the logic it condones. The use of impredicative methods in classical mathematics is a site of traditional intuitionistic attacks. I undertake an examination of what the anti-realist attitude towards such methods should be. This question is of interest both because such methods are deeply implicated in classical mathematical theory of analysis and because intuitionistic semantic theories make use of impredicative methods. I attempt to construct the outlines of a set theory which is anti-realistically acceptable but which, although having no antecedent repugnance for impredicative methods as such, appears to be too weak to offer an anti-realistic vindication of impredicative methods in general. I attempt to exonerate intuitionistic semantic theories in their use of impredicative methods by showing that a partial order relying on the nature of our grasp of the intuitionistic meaning stipulations for the logical constants precludes a possible circularity.

M.Sc. (Mathematics) / This dissertation describes the solution toa specific logical problem, the satisfiability problem, in a logic of games called Alternating-time Temporal Logic (ATL). Computation Tree Logic (CTL) is a discrete branching-time temporal logic for reasoning about labelled transition systems. ATL extends CTL to describe gametheoretic situations, where multiple agents together determine the evolution of the system. In particular, ATL explicitly provides for describing the abilities of coalitions of agents in such systems. Weprovide an automata-based decision procedure for ATL by translating the satisfiability problem for an ATL formula to the nonemptiness problem for an Alternating Biichi 'free Automaton. The key result that enables this translation is a oundedbranching tree model theorem for ATL, proving that a satisfiable formula is also satisfiable in a tree model of bounded branching degree. In terms of complexity, we show that satisfiability in ATL is complete for exponential time, which agrees with the corresponding complexity result for the fragment CTL. Closely related to ATL is an independently developed family of modal logics, the Coalition Logics. The presented results also provide a satisfiability procedure for Extended Coalition Logic interpreted over strongly playable coalition models. The structure of the dissertation is as follows: • Chapter 1 is an introduction to the topic, provides an overview of the results and a preview of the dissertation. • Chapter 2 presents some mathematical preliminaries regarding trees, automata, fixed points and game theory. • Chapter 3 discusses CTL and in particular an automata-based satisfiability procedure for CTL. • Chapter 4 introduces Alternating-time Temporal Logic (ATL) as a logic of games. • Chapter 5 contains the main results of the dissertation: first we prove a boundedbranching tree model property for ATL. Then the construction of the required automaton for satisfiability checking is described. • Chapter 6 relates the present work to some other logics of games, and in particular the Coalition Logics. • Chapter 7 finalises the dissertation with a conclusion and a look at some future research directions that might be pursued following the present work.

M.Sc. / This short dissertation is intended to give a brief account of the history and current state of affairs in the field of study called 'Zero-one Laws'. The probability of a property P on a class of finite relational structures is defined to be the limit of the sequence of fractions, of the n element structures that satisfy the property P, as n tends to infinity. A class of properties is said to have a Zero-One law if the above limit, which is usually called the asymptotic probability of the property with respect to the given class of finite structures, is either 0 or 1 for each property. The connection to the field of Mathematical Logic is given by the surprising fact that the class of properties definable by a first-order sentence has a Zero-One law with respect to the class of all finite relational structures of the common signature. We cover this result in more detail and discuss several further Zero- One laws for higher-order logics. In particular we will be interested in all those modal formulae which are 'almost surely' frame valid in the finite, i.e. those which have an asymptotic probability equal to 1 with respect to the class of all finite frames. Our goal is to find a purely logical characterization of these formulae by finding a set of axioms which describe such modal formulae absolutely. We devise a strategy and provide some Java programs to aid in this search for future research

We begin with a disucssion of some of the serious deficiencies of first order predicate languages. These include the non-characterizability by first order sentences of such common mathematical structures as the class of well-ordered sets, the class of finite sets, the class of Archimedean fields and the standard models of arithmetic and analysis. Two methods of generalizing first order predicate languages are then studied. The first approach is to allow for "expressions of infinite length"; the second method is the introduction of "generalized quantifiers." For the languages resulting from each approach, we consider to what extent such deficiencies as those mentioned above may be overcome and also to what extent some of the elementary model-theoretic and proof-theoretic theorems of first order logic may be generalized to these new languages. Among the languages with expressions of infinite length, we first consider the Lω₁ω languages which generalize first order languages by extending the recursive definition of a formula to allow countable conjunctions and disjunctions of formulas as formulas. It is shown that with the use of such languages we are able to describe categorically the standard model of arithmetic, the class of finite sets, the class of Archimedean fields and other common mathematical structures which cannot be characterized in first order languages. Generalizations of the Lowenheim-Skolem and completeness theorems of first order logic are given as well as a countable isomorphism theorem due to Dana Scott. We make use of a characterization of rank equivalence due to Carol Karp to demonstrate that neither the standard model of analysis nor the class of well-ordered sets may be described in any Lω₁ω -language. In fact, our argument indicates that these characterizations are not possible in any extension of a Lω₁ω - language which, for any infinite cardinal α , allows as formulas conjunctions and disjunctions of less than a formulas. This result leads us naturally to a consideration of the class of Lω₁ω - languages, any element of which is obtained from a Lω₁ω - language by modifying the rules for formula formation to allow not only denumerable conjunctions and disjunctions but also quantifications over denumerable sets of variables. (These ideas are made more precise in the text of the thesis.) The standard model of analysis and the class of well-ordered sets are each seen to be characterizable by single Lω₁ω - sentences. Other infinitary languages are also mentioned, including languages with infinitely long atomic formulas. Among the languages with generalized quantifiers we restrict ourselves to the L(Qα) - languages, where α is an ordinal, which are obtained from first order languages by adding a new quantifier symbol Qα to be read "there exist Ɲα... .” In addition to being able to characterize sets of various cardinalities, we give a categorical description of the standard model of arithmetic by a single L(Qօ) - sentence. Among the model-theoretic results possible are generalizations of the compactness theorem, Lŏs's theorem and the downward Lowenheim - Skolem theorem of first order logic. Finally, on the proof-theoretic side, we show that in the case α = 0 there exists no recursive axiomatization which yields a completeness result; in the case α = 1 , however, such an axiomatization is possible. / Science, Faculty of / Mathematics, Department of / Graduate

This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.