Predicative topos theory and models for constructive set theory

Abstract

"Predicative topos theory and models for constructive set theory" is a PhD thesis written at the University of Utrecht under supervision of prof. Ieke Moerdijk. It concerns a categorical analysis of predicative and constructive formal systems, especially Peter Aczel's constructive Zermelo-Fraenkel set theory (CZF).

CZF provides an adequate framework for developing constructive mathematics, so that a mathematical study of CZF would provide direct insight in the nature of constructive mathematics. Moreover, Aczel managed to interpret CZF in Martin-Löf's constructive type theory. This vindicates directly the constructive nature of the set theory CZF.

CZF has been around since 1978, but has recently again attracted the attention of some proof-theorists. But now more semantic approaches remain underdeveloped. The thesis was meant to remedy this state of affairs, building on earlier work by Moerdijk and Palmgren. Their idea was develop a semantics for CZF based on category theory and algebraic set theory.

The study of the set theory IZF, which can be considered as the impredicative counterpart of CZF, has been greatly helped by the subject called topos theory. Topos theory is a rich subject with plenty of implications for the set theory IZF.

The thesis shows that in a similar manner CZF profits from a notion of predicative topos. The notion of a topos is too impredicative for studying CZF, but a ``predicativised'' version should be very helpful. In their papers, Moerdijk and Palmgren discuss the notion of a PiW-pretopos which the thesis argues is a suitable predicative analogue to the notion of a topos.

The first chapters of the thesis continue along these lines. After an introduction to the field in the first chapter, the second chapter defines the notion of a PiW-pretopos. Special care is taken to formulate the notion of a W-type, an essential ingredient of the notion of a PiW-pretopos, for which the thesis proves a new characterisation result which helps to recognise W-types in concrete cases. The third chapter discusses two new closure properties of PiW-pretoposes.

The fourth chapter makes the connection to Aczel's set theory CZF. Following Moerdijk and Palmgren, the thesis discusses how using the tools of algebraic set theory, models of CZF can be constructed inside PiW-pretoposes. It shows how a model discovered independently by Streicher and Lubarsky can be understood in these terms. Firstly, it shows that the two models are the same, and secondly it shows which constructive principles hold in the model.

In the last two chapters of the thesis, which are joint work with dr. Federico De Marchi, discuss non-well-founded structures. Non-well-founded analogues of W-types, called M-types, are discussed in the fifth chapter of the thesis. In the sixth and final chapter, the thesis proves a final coalgebra theorem in an abstract categorical context. This is then shown to provide models for non-well-founded versions of set theories like the classical ZFC, but also IZF or CZF, in the setting of algebraic set theory.