Let ZFB be ZF + “every set is the same size as a wellfounded set”. Then the following are true.

Every sentence true in every (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e., ZFB is the theory of Rieger-Bernays permutation models of models of ZF)

ZF and ZFAFA are both extensions of ZFB conservative for stratified formul{\ae}.

{The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.

For $\al$ less than $ \eo$ let $N\al$ be the number of occurrences of $\om$ in the Cantor normal form of $\al$. Further let $\lh n$ denote the binary length of a natural number $n$, let $\lhh n$ denote the $h$-times iterated binary length of $n$ and let $\inv n$ be the least $h$ such that $\lhh n \leq2$. We show that for any natural number $h$ first order Peano arithmetic, $\PA$, does not prove the following sentence: For all $ K$ there exists an $ M$ which bounds the lengths $n$ of all strictly descending sequences $\langle \al₀,... ,\aln\rangle$ of ordinals less than $\eo$ which satisfy the condition that the Norm $N\al_i$ of the $i$-th term $\al_i$ is bounded by $K + \lh i\cdot \lhh{i}$.

As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, $\PRA$, proves that for all $K$ there is an $M$ which bounds the length $n$ of any strictly descending sequence $\langle \al₀,... ,\aln\rangle $ of ordinals less than $\eo$ which satisfies the condition that the Norm $N\al_i$ of the $i$-th term $\al_i$ is bounded by $K + \lh i \cdot \inv i$. The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations.

Using results from Otter and from Matou{\v{s}}ek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856... .

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta₁$. We show that $I\Delta₁$ is independent from the set of all true arithmetical $\Pi₂$-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta₁$-induction.

An open problem formulated by J. Paris (see \cite{CloKra,HP}) is whether $I\Delta₁$ proves the corresponding least element principle for decidable predicates, $L\Delta₁$ (or, equivalently, the $\Sigma₁$-collection principle $B\Sigma₁$). We reduce this question to a purely computation-theoretic one.

Let $(M,\leq,...)$ denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of $M$ determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

In “Bounding minimal degrees by computably enumerable degrees” by A. Li and D. Yang, (this Journal, \cite{LY}), the authors prove that there exist non-computable computably enumerable degrees c > a > z such that any minimal degree m being below c is also below a. We analyze the proof of their result and show that the proof contains a mistake. Instead we give a proof for the opposite result.

This paper developed from Shelah’s proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

Dual combinators emerge from the aim of assigning formulas containing $\leftarrow$ as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for $Z$-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

Kaneko-Suzuki developed epistemic logics of shallow depths with multiple players for investigations of game theoretical problems. By shallow depth, we mean that nested occurrences of belief operators of players in formulae are restricted, typically to be of finite depths, by a given epistemic structure. In this paper, we develop various methods of surgical operations (cut and paste) of epistemic world models. An example is a bouquet-making, i.e., tying several models into a bouquet. Another example is to engraft a model to some branches of another model. By these methods, we obtain various meta-theorems on semantics and syntax on epistemic logics. To illustrate possible uses of our meta-theorems, we present one game theoretical theorem, which is also a meta-theorem in the sense of logic.

Given well ordered countable sets of the form $\lamphi$, we consider Borel mappings from $\lamphiom$ with countable image inside the ordinals. The ordinals and $\lamphiom$ are respectively equipped with the discrete topology and the product of the discrete topology on $\lamphi$. The Steel well-ordering on such mappings is defined by $\phi\minf\psi$ iff there exists a continuous function $f$ such that $\phi(x)\leq\psi\circ f(x)$ holds for any $x\in\lamphiom$. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal $\alpha$, a mapping $\T{\alpha}$ whose rank is precisely $\alpha$ in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by $\alpha$. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

We study the degree of (weak) $Q$-pointness, and that of (weak) $P$-pointness, of ideals on a regular infinite cardinal.

It is shown that the automorphism group of a relation algebra $\mla P$ constructed from a projective geometry $P$ is isomorphic to the collineation group of $P$. Also, the base automorphism group of a representation of $\mla P$ over an affine geometry $D$ is isomorphic to the quotient of the collineation group of $D$ by the dilatation subgroup. Consequently, the total number of inequivalent representations of $\mla P$, for finite geometries $P$, is the sum of the numbers \[ \frac{|\aut P|}{|\aut {D}|\mathbin{/}|\dil{D}|}, \] where $D$ ranges over a list of the non-isomorphic affine geometries having $P$ as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

In \cite{MessmerW1995} Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant $e$ equipped with a (certain) \Hde. We propose a variant of their theory, using a sequence of $e$ commuting \Hds. In contrast to \cite{MessmerW1995} our \Hds are iterative.

We settle a number of questions concerning definability in first order logic with an extra predicate symbol ranging over semi-linear sets. We give new results both on the positive and negative side: we show that in first-order logic one cannot query a semi-linear set as to whether or not it contains a line, or whether or not it contains the line segment between two given points. However, we show that some of these queries become definable if one makes small restrictions on the semi-linear sets considered.