A Reflection Principle for Potential Infinite Models of Type Theory

Abstract

Denotational models of type theory, such as set-theoretic, domain-theoretic, or category-theoretic models use (actual) infinite sets of objects in one way or another. The potential infinite, seen as an extensible finite, requires a dynamic understanding of the infinite sets of objects. It follows that the type $nat$ cannot be interpreted as a set of all natural numbers, $\lbrack\!\lbrack nat \rbrack\!\rbrack = \mathbb{N}$, but as an increasing family of finite sets $\mathbb{N}_i = \{0, \dots, i-1\}$. Any reference to $\lbrack\!\lbrack nat \rbrack\!\rbrack$, either by the formal syntax or by meta-level concepts, must be a reference to a (sufficiently large) set $\mathbb{N}_i$. We present the basic concepts for interpreting a fragment of the simply typed $\lambda$-calculus within such a dynamic model. A type $\varrho$ is thereby interpreted as a process, which is formally a factor system together with a limit of it. A factor system is very similar to a direct or an inverse system, and its limit is also defined by a universal property. It is crucial to recognize that a limit is not necessarily an unreachable end beyond the process. Rather, it can be regarded as an intermediate state within the factor system, which can still be extended. The logical type $bool$ plays an important role, which we interpret classically as the set $\{true, false\}$. We provide an interpretation of simply typed $\lambda$-terms in these factor systems and limits. The main result is a reflection principle, which states that an element in the limit has a ``full representative'' at a sufficiently large stage within the factor system. For propositions, that is, terms of type $bool$, this implies that statements about the limit are true if and only if they are true at that sufficiently large stage.