Generalizing classical and effective model theory in theories of operations and classes

Abstract

Mancosu, P., Generalizing classical and effective model theory in theories of operations and classes, Annas of Pure and Applied Logic 52 (1991) 249-308. In this paper I propose a family of theories of operations and classes with the aim of developing abstract versions of model-theoretic results. The systems are closely related to those introduced and already used by Feferman for developing his program of ‘explicit mathematics’. The theories in question are two-sorted, with one kind of variable for individuals and the other for classes. The individual variables range over a domain closed under pairing and containing, among other things, natural numbers and (partial) operations.

All the theories used assume a common group of axioms that insure that the individuals in the domain satisfy the conditions for a partial combinatory (applicative) algebra with pairing. We also assume a class $\text{N}$ of natural numbers and a class induction axiom on $\text{N}$. Finally there are various class existence axioms. I work mainly with three theories, FMT₀, FMT and FMT_Ω, which differ from each other both in their class existence axioms as well as in some further applicative axioms.

These theories have various interpretations in which every object is explicitly presented by some means of definition as well as classical, or ‘standard’, set-theoretical interpretation. The systems are formulated in a flexible general language which can be used to prove abstract theorems of mathematics (analysis, algebra, model theory, etc.), thereby (depending on the system) generalizing recursive, hyperarithmetic and admissible versions of classical mathematics.

The aim of my work is to give an abstract development of portions of model theory in the afore-mentioned theories, in a way to look as much as possible like classical mathematics. Thus, FMT₀ generalizes portions of countable and recursive model theory; FMT further generalizes portions of countable and hyperarithmetical model theory and finally FMT_Ω provides a generalization of the classical L(Q)-completeness theorem and of an admissible version of L(Q)-completeness due to Bruce and Keisler. This continues the development of the program mentioned above, originating with Feferman.