Choice sequences and knowledge states: extending the notion of finite information to produce a clearer foundation for intuitionistic analysis

Abstract

There are currently four major formal foundational systems for intuitionistic analysis: LS, CS (both in Troelstra 1977), FIM (Kleene and Vesley 1965) and the derivable FIRM-INT (Moschovakis 2016). All of these systems rely on different universes of choice sequences and different conceptions of what a choice sequence is. There is a strong common ground between these systems as they use the same very restrictive notion of finite information when dealing with these choice sequences { the notion of restricting ourselves to initial segments.
This text extends the notion of a choice sequence given in Fletcher (1998) and uses it to construct a generalised system capable of expressing results about intensional properties of choice sequences. This is achieved by constructing a language capable of representing intensional first order restrictions on choice sequences (the language of knowledge states) and their relations to other sequences. This extended system allows us to formulate a notion of lawlessness that evades a series of paradoxes highlighted in Fletcher (1998), allows us to prove a generalised form of open data and offers additional clarity to other key areas of the theory. When a certain set of restrictions are applied to this extended theory (extensionality and a second order restriction on knowledge states) we obtain a system suitable for the foundation of analysis.