Episodes in Model-Theoretic Xenology: Rationals as Positive Integers in R#

Meyer and Mortensen’s Alien Intruder Theorem includes the extraor- dinary observation that the rationals can be extended to a model of the relevant arithmetic R♯, thereby serving as integers themselves. Al- though the mysteriousness of this observation is acknowledged, little is done to explain why such rationals-as-integers exist or how they operate. In this paper, we show that Meyer and Mortensen’s models can be identified with a class of ultraproducts of finite models of R♯, providing insights into some of the more mysterious phenomena of the rational models.

Independence-friendly logic without Henkin quantification

We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular $$\mathrm {IF}$$ IF sentences, this amounts to the study of the fragment of $$\mathrm {IF}$$ IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.

Undecidability of the Logic of Partial Quasiary Predicates

We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $\varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $\varPi ^0_1$-complete—over finite structures.

Axiomatizing Rectangular Grids with no Extra Non-unary Relations

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n); s(n) = Ω(log(n)).

Finite spaces and an axiomatization of the Lefschetz number

In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite {T_{0}} -spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite {T_{0}} -spaces. However, using the notion of the subdivision of a finite {T_{0}} -space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle {\mathbb{S}^{1}} .