The complexity of classification problems for models of arithmetic

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.Comment: 15 page

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

A standard model of Peano arithmetic with no conservative elementary extension

AbstractThe principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion NA≔(N,A)A∈A of the standard model N≔(ω,+,×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension NA∗=(ω∗,…) of NA, there is a subset of ω∗ that is parametrically definable in NA∗ but whose intersection with ω is not a member of A. We also establish other results that highlight the role of countability in the model theory of arithmetic.Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/FIN (where FIN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing

Set theoretical analogues of the Barwise-Schlipf theorem

We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus $\Delta^1_1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to Goedel-Bernays class theory plus $\Sigma^1_1$-Choice.Comment: 16 pages. This version corrects some minor typos in the previous draft. The preliminaries section of the paper has a substantial text overlap with arXiv:1910.0402

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).Comment: 17 pages, 1 figur

Truth, collection and deflationism in models of peano arithmetic

This thesis focuses on adding collection axioms to satisfaction classes and exploring the suitability of a formal deflationary truth predicate. Chapter 2 proves that every nonstandard, recursively saturated model of PA has a satisfaction class in which all collection axioms are true. Chapter 3 explores collection axioms for the language with the satisfaction predicate, ℒS, and proves that these entail the theory of chapter 2. This chapter then demonstrates a method of closing a model with a satisfaction class to produce a new model with an induced satisfaction class, which it is conjectured will not satisfy all Ʃ1 collection axioms in ℒS. In chapter 4 we conjecture that a new formulation of Visser and Enayat's construction of extensions of models with a satisfaction classes [5] will provide elementary extensions. Using this conjecture, we demonstrate new Tarski axioms provide satisfaction classes with Ʃ1 collection axioms and that these axioms can be built into the theory by reducing the language to one where formulas are stratified. Finally, in chapter 5 we argue for a new definition of a deflationary truth predicate and show that this entails there are no formalisations of a deflationary truth predicate for the full nonstandard language of arithmetic

Existential witness extraction in classical realizability and via a negative translation

We show how to extract existential witnesses from classical proofs using Krivine's classical realizability---where classical proofs are interpreted as lambda-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Sigma01-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS), 201

Infinity and Continuum in the Alternative Set Theory

Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vop\v enka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. This incidentally provides a natural solution to some classic philosophical problems such as the composition of a continuum, Zeno's paradoxes and sorites. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vop\v enka's theory reverses the process: he models the finite in the infinite.Comment: 25 page