In these notes we go through the very basic results from descriptive set theory. In our approach to the topic, the use of games is more explicit than what is usual in the literature. We do not assume any previous knowledge on set theory nor on first-order logic. For the history of the topic, see [Mo] (Borel* sets are due to D. Blackwell and Vaught codes are due to R. Vaught). For further readin...

In this paper we explore the connections between the monadic second-order theory of one successor (MSO[<] for short) and the theories of ω-layered structures for time granularity. We first prove that the decision problem for MSO[<] and that for a suitable first-order theory of the upward unbounded layered structure are inter-reducible. Then, we show that a similar result holds for suitable chai...

Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order senten...

Both second order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first or...

We present new parameterized preservation properties that provide for each natural number k, semantic characterizations of the ∃∀ and ∀∃ prefix classes of first order logic sentences, over the class of all structures and for arbitrary finite vocabularies. These properties, that we call preservation under substructures modulo k-cruxes and preservation under k-ary covered extensions respectively,...

Work on how to axiomatize the subtheories of first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.

Type classes and overloading are shown to be independent concepts that can both be added to simple higher-order logics in the tradition of Church and Gordon, without demanding more logical expressiveness. In particular, model-theoretic issues are not affected. Our metalogical results may serve as a foundation of systems like Isabelle/Pure that offer the user Haskell-style order-sorted polymorph...

Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples are theories of vector spaces and algebraically closed fields; also the degenerate example of the theory of infinite ‘pure’ sets where the only structure comes from equality. Algebraic closure in a str...

Eilenberg, Elgot and Shepherdson showed in 1969, [9], that a relation on finite words over a finite, non-unary alphabet with p letters is definable in the first order logic with p + 2 predicates for the relations equal length, prefix and last letter is a (for each letter a ∈ Σ) if and only if it can be recognized by a finite multitape synchronous automaton, i.e., one whose read heads move simul...

This paper proposes a notion of finitely-verifiable classes of sentences. Informally, a class of sentences is finitely-verifiable if whether a sentence in this class is a theorem of a given theory can be checked with respect to a finite set of models of the theory. The usefulness of this notion is illustrated using examples from arithmetics, first-order logic, game theory, and planning.