This collection of articles on Models of Arithmetic is dedicated to the memory of Zygmunt Ratajczyk, who contributed a number of important results to the field, and who unexpectedly died in February 1994.

Zygmunt Ratajczyk was a deep and subtle mathematician who, with mastery, used sophisticated and technically complex methods, in particular combinatorial and proof-theoretic ones. Walking always along his own paths and being immune from actual trends and fashions he hesitated to publish his results, looking endlessly for their improvement.

We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.

The paper presents four open problems. One concerns a possible converse to Tarski's undefinability of truth theorem, and is of a general character. The other three are more specific. The questions are about some special $\omega_1$-like models, initial segments of countable recursively saturated models of PA, and about extendability of automorphisms. In each case a partial answer is given. All partial solutions are based on applications of inductive satisfaction classes.

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

A model $(M,<,\ldots)$ is said to be $\kappa$-like if $\mbox{card $\;$} M =\kappa$ but for all $a\in M$, $\mbox{card}\{x \in M: x< a\}< \kappa$. In this paper, we shall study sentences true in $\kappa$-like models of arithmetic, especially in the cases when $\kappa$ is singular. In particular, we identify axiom schemes true in such models which are particularly `natural' from a combinatorial or model-theoretic point of view and investigate the properties of models of these schemes.

The theory PA(aa), which is Peano Arithmetic in the context of stationary logic, is shown to be consistent. Moreover, the first-order theory of the class of finitely determinate models of PA(aa) is characterized.

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey's Theorem for Pairs is not strong enough to prove $ACA_0$, the comprehension scheme for arithmetical formulas, within the base theory $RCA_0$, the comprehension scheme for recursive formulas. We also show that Ramsey's Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within $RCA_0$. In particular, Ramsey's Theorem for Pairs is not conservative over $RCA_0$ for $\Pi^0_4$-sentences.

A new concept of model for the US-tense logic is introduced, in which ternary relations of betweenness are adjoined to the usual early-later relation. The class of these new models, which contains the class of Kripke models, satisfies, contrary to that, the Distinguishable Model Theorem, in the sense that each model is equivalent to a model in which no two points verify exactly the same formulas.

In this paper we give an axiom system of a logic which we call an approximation logic (AL), whose Lindenbaum-Tarski algebra is a strong Bunge algebra (or simply s-Bunge algebra), and show that

For every s-Bunge algebra $B$, a quotient algebra $B^*$ by a maximal filter is isomorphic to the simplest nontrivial s-Bunge algebra $\Omega=\{0,a,1\}$;

The Lindenbaum algebra of AL is an s-Bunge algebra;

AL is complete;

AL is decidable.

In this paper we describe, in a purely algebraic language, truth-complete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.