We propose a proof of the downward Löwenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms. This proof is simpler, and easily understood by the students, although it requires, when defining the semantics of first-order logic to introduce first a few notions inherited from game theory such as the one of an evaluation game. 1998 ACM Subject Cla...

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result – which shows that the Excluded Middle principle can be used to eliminate Skolem functions – has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Inter...

We show that a strong form of the so called Lindstrr om's Theorem 4] fails to generalize to extensions of L ! and L : For weakly compact there is no strongest extension of L ! with the (;)-compactness property and the LL owenheim-Skolem theorem down to. With an additional set-theoretic assumption, there is no strongest extension of L with the (;)-compactness property and the LL owenheim-Skolem ...

Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the environment, which ma...

Introduction 2 • PTIME decidable uniform word problems for quasi-varieties • CS applications: type inference systems, program analysis, decision procedures in ATP • fundamental effective methods for establishing PTIME upper bounds • local theories (Givan, McAllester 92) capture PTIME • algebraic criteria by Skolem (1920), Evans (1951), and Burris (1995) • relation between these criteria Introdu...