A Note on Computable Embeddings for Ordinals and Their Reverses

We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that although $\{\omega \cdot 2, \omega^\star \cdot 2\}$ is computably embeddable in $\{\omega^2, {(\omega^2)}^\star\}$, the class $\{\omega \cdot k,\omega^\star \cdot k\}$ is \emph{not} computably embeddable in $\{\omega^2, {(\omega^2)}^\star\}$ for any natural number $k \geq 3$.Comment: 13 pages, accepted to CiE 202

The model-theoretic complexity of automatic linear orders

Automatic structures are—possibly infinite—structures which are finitely presentable by means of finite automata on strings or trees. Largely motivated by the fact that their first-order theories are uniformly decidable, automatic structures gained a lot of attention in the "logic in computer science" community during the last fifteen years. This thesis studies the model-theoretic complexity of automatic linear orders in terms of two complexity measures: the finite-condensation rank and the Ramsey degree. The former is an ordinal which indicates how far a linear order is away from being dense. The corresponding main results establish optimal upper bounds on this rank with respect to several notions of automaticity. The Ramsey degree measures the model-theoretic complexity of well-orders by means of the partition relations studied in combinatorial set theory. This concept is investigated in a purely set-theoretic setting as well as in the context of automatic structures.Auch im Buchhandel erhältlich: The model-theoretic complexity of automatic linear orders / Martin Huschenbett Ilmenau : Univ.-Verl. Ilmenau, 2016. - xiii, 228 Seiten ISBN 978-3-86360-127-0 Preis (Druckausgabe): 16,50

Constructions with Countable Subshifts of Finite Type

We present constructions of countable two-dimensional subshifts of finite type (SFTs) with interesting properties. Our main focus is on properties of the topological derivatives and subpattern posets of these objects. We present a countable SFT whose iterated derivatives are maximally complex from the computational point of view, constructions of countable SFTs with high Cantor-Bendixson ranks, a countable SFT whose subpattern poset contains an infinite descending chain and a countable SFT whose subpattern poset contains all finite posets. When possible, we make these constructions deterministic, and ensure the sets of rows are very simple as one-dimensional subshifts

SEPARATING FRAGMENTS OF WLEM, LPO, AND MP

Abstract. We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov's Principle, including Weak Markov's Principle, do not imply each other. §1. Introduction. At the beginning of the twentieth century, Brouwer identified a number of constructively dubious principles, which Bishop later, in his 1967 monograph Omniscience principles are commonly used to show the independence of more subject specific theorems: if a (classical) result constructively implies an omniscience principle, then it cannot be proved using constructive techniques. By separating different omniscience principles over IZF we make this task easier: if under the assumption of a classical result together with an omniscience principle we can derive a stronger omniscience principle, then we can still conclude that the classical theorem is nonconstructive. More generally, implications among these principles and theorems of mainstream mathematics have been studied for a long time. Often this is the motivation for introducing these principles (some references being provided with the principles below), and often this study is done for foundational reasons after the principles are already established (as, for instance, in In this paper we present many models, often related to each other, that separate a large number of the omniscience principles defined in terms of binary sequences and related principles. The genesis of this work was the first author's question to the second of whether Richman's LLPO n hierarchy [23] could be separated, a question about results. Since then, much interest has shifted to technique: could an argumen

An Introduction to Set Theory and Topology

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. Examples are taken now and then from analysis, but they are not logically necessary for the development of the material. The only real prerequisite is the level of mathematical interest, maturity and patience needed to handle abstract ideas and to read and write careful proofs. A few very capable students have taken this course before introductory analysis (even, rarely, outstanding university freshmen) and invariably they have commented later on how material eased their way into analysis.https://openscholarship.wustl.edu/books/1020/thumbnail.jp

Axiomatization and Incompleteness in Arithmetic and Set Theory

Axiomatization and Incompleteness in Arithmetic and Set Theory Wesley Duncan Wrigley I argue that are (at least) two distinct kinds of mathematical incompleteness. Part A of the thesis discusses Gödelian incompleteness, while Part B is concerned with set-theoretic incompleteness. Both parts are concerned with the philosophical justification of reflection principles and other axiomatic devices which can be used to reduce incompleteness, and in particular with the justification of such devices from the philosophical standpoint of Kurt Gödel. In Part A I consider Gödel's disjunctive argument. In chapter 1, I argue that the non-mechanical mind considered by Gödel is best modelled by a theory constructed using the transfinite iterated application of a soundness reflection principle to PA. I argue that Feferman's completeness theorem shows this account of the mind to be incompatible with some elementary assumptions in the epistemology of arithmetic. In chapter 2, these considerations are developed into a positive argument for the existence of absolutely undecidable arithmetical propositions. The consequences for the indefinite extensibility of the concept natural number are then discussed. I argue that properly understood, Feferman's theorem refutes Dummett's position in the debate. I begin part Part B in chapter 3, by reconstructing a version of Gödel's platonism, called conceptual platonism. I then examine how such a position relates to various means of reducing set-theoretic incompleteness. In chapter 4 I argue that there is some prospect for this position of effecting a limited reduction in incompleteness by means of reflection principles justified by mathematical intuition. However, such priniciples are incompatible with Gödel's commitment to platonism about properties of properties of sets. In chapter 5 I argue that conceptual platonism does not lend support to the view that a substantial reduction in incompleteness can be effected by large cardinal axioms justified using extrinsic methods analogous to the principles of theory choice in natural science. This undercuts the traditional justification for many large cardinal axioms, so I end with a sketch of how conceptual platonism could be modified to rehabilitate the large cardinals program.This thesis was generously funded by an Arts and Humanities Research Council Doctoral Scholarship