Set Theory Without the Axiom of Foundation

Declaration I declare that this essay is work done as part of the Part III Examination. It is the result of my own work, and except where stated otherwise, includes nothing which was performed in collaboration. No part of this essay has been submitted for a degree or any such qualification. Self-reference has been a concept which has concerned philosophers for millennia, and which gives rise to many paradoxes. A classic example of such a paradox would be a strip of paper with 'the statement on the other side of this strip of paper is true' on the front, and 'the statement on the other side of this strip of paper is false' on the back 1 where a contradiction is brought about by each statement referring to the other. Despite the paradoxes which may arise if self-reference occurs freely, the use of self-referential structures is widespread, appearing in branches of computer science, artificial intelligence, situation semantics and linguistics, to name but a few. The concept of self-reference is easily understood and self-reference is a natural way to describe certain systems and phenomena. Heuristically, however, one may use the self-referentiality of the structure to say that the limit is a solution to x = 1 + 1 x , i.e. x 2 − x − 1 = 0, and this is simply solved to give x = 1+ √ 5 2 ; it cannot be the other root as the desired limit is clearly non-negative. 1 [7] suggests that it is possible to solve this paradox by glueing the ends of the strip together with a half-twist, rendering any assertions about the other side of the strip nonsensical. 1 1 INTRODUCTION 1.1 Motivation Also, it would be immediately clear to most mathematicians what object the symbols are intended to represent. In the widely-accepted Zermelo–Fraenkel (ZF) set theory such a construction is forbidden by the foundation axiom and yet the concept it expresses is simple to understand and not obviously contradictory. A finite list is often constructed as the ordered pair of its head and its tail; infinite lists (or streams) are also of importance, yet generalising the usual construction of a finite list to the infinite case is forbidden in the wellfounded world. While there are alternatives to the notion of a finite list which do generalise (and also notions of ordered pair which permit wellfounded infinitary constructions) it would …