Infinite Time Computable Model Theory

Abstract

We introduce infinite time computable model theory, the com- putable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of zfc. Computable model theory is model theory with a view to the computability of the structures and theories that arise (for a standard reference, see (EGNR98)). Infinite time computable model theory, which we introduce here, carries out this program with the infinitary notions of computability provided by infinite time Turing ma- chines. The motivation for a broader context is that, while finite time computable model theory is necessarily limited to countable models and theories, the infinitary context naturally allows for uncountable models and theories, while retaining the computational nature of the undertaking. Many constructions generalize from finite time computable model theory, with structures built on N, to the infinitary theory, with structures built on R. In this article, we introduce the basic theory and con- sider the infinitary analogues of the completeness theorem, the Lowenheim-Skolem Theorem, Myhill's theorem and others. It turns out that, when stated in their fully general infinitary forms, several of these fundamental questions are independent of zfc. The analysis makes use of techniques both from computability theory and set theory. This article follows up (Ham05). 1.1. Infinite time Turing machines. The definitive introduction to infinite time Turing machines appears in (HL00), but let us quickly describe how they work. The