The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm {CT}_0$). Furthermore, we (...) extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$. (shrink)

We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic ${\textsf {(PA)}}$. More specifically, let Elementary Arithmetic ${\textsf {(EA)}}$ be the fragment $\mathsf {I}\Delta _0 + \mathsf {Exp}$ of ${\textsf {PA}}$, and let ${\textsf {CT}}^-[{\textsf {EA}}]$ be the extension of ${\textsf {EA}}$ by the commonly studied axioms of compositional truth ${\textsf {CT}}^-$. We investigate both local and global properties of the family of first order theories of the form ${\textsf {CT}}^-[{\textsf {EA}}] +\alpha $, where $\alpha (...) $ is a particular way of expressing “ ${\textsf {PA}}$ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to ${\textsf {PA}}$ and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of ${\textsf {PA}}$. (shrink)

We introduceself-divisibleultrafilters, which we prove to be precisely those$w$such that the weak congruence relation$\equiv _w$introduced by Šobot is an equivalence relation on$\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that$w$is self-divisible if and only if$\equiv _w$coincides with the strong congruence relation$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$is a profinite group. We also construct an ultrafilter$w$such that$\equiv _w$fails to be symmetric, and describe the interaction between the (...) aforementioned quotient and the profinite completion$\hat {{\mathbb Z}}$of the integers. (shrink)

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation (...) in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic. (shrink)

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we (...) show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T. (shrink)

We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

By suitably adapting an argument of Hirschfeld , we show that there is a single Δ1 formula that defeats “bounded collection” for any model of II2 Arithmetic that is either a recursive ultrapower or an existentially complete model. Some related facts are noted. MSC: 03F30, 03C62.

In this paper, we show that equation image is a equation image-conservative extension of BΣ1 + exp, thus it does not imply IΣ1.

We show that for every countable recursively saturated model M of Peano arithmetic and every subset A⊆M, there exists a full satisfaction class SA⊆M2 such that A is definable in (M,SA) without parameters. It follows that in every such model, there exists a full satisfaction class which makes every element definable, and thus the expanded model is minimal and rigid. On the other hand, as observed by Roman Kossak, for every full satisfaction class S there are two elements which have (...) the same arithmetical type, but exactly one of them is in S. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of the first result proved here for full satisfaction classes was obtained also by Roman Kossak for partial inductive satisfaction classes. However, the proof relied on the induction scheme in a crucial way, so recapturing the result in the setting of full satisfaction classes requires quite different arguments. (shrink)

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

In this paper we will show that for every cutIof any countable nonstandard model$\mathcal {M}$of$\mathrm {I}\Sigma _{1}$, eachI-small$\Sigma _{1}$-elementary submodel of$\mathcal {M}$is of the form of the set of fixed points of some proper initial self-embedding of$\mathcal {M}$iffIis a strong cut of$\mathcal {M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model$\mathcal {M}$of$ \mathrm {I}\Sigma _{1} $. In addition, we will find some criteria for extendability of initial (...) self-embeddings of countable nonstandard models of$ \mathrm {I}\Sigma _{1} $to larger models. (shrink)

Fix a countable nonstandard model M of Peano arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions N≻M that are allowed, we still find that there are 2ℵ0 possible theories of (N,M) for such N’s.

A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including $\mathsf {PA}$ [39], $\mathsf {ZF}$, $\mathsf {Z}_2$, and $\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of $\mathsf {Z}_2$ and $\mathsf {KM}$ gives non-tight theories. Specifically, we show that $\mathsf {GB}$ and $\mathsf {ACA}_0$ each admit different bi-interpretable extensions, (...) and the same holds for their extensions by adding $\Sigma ^1_k$ -Comprehension, for $k \ge 1$. These results provide evidence that tightness characterizes $\mathsf {Z}_2$ and $\mathsf {KM}$ in a minimal way. (shrink)

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction (...) tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations. (shrink)

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction (...) tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations. (shrink)