دانلود کتاب Non-Well-Founded Sets

ForewordTo my way of thinking, mathemtical logic is a branch of appliedmathematics. It applies mathematics to model and study varioussorts of symbolic systems: axioms, proofs, programs, computers,or people talking and reasoning together. This is the only view ofmathematical logic which does justice to the logician's intuitionthat logic really is a field, not just the union of several unrelatedfields. One expects that logic, 88 a branch of applied mathematics,will not only use existing tools from mathematics, but also thatit will lead to the creation of new mathematical tools, tools thatarise out of the need to model some real world phenomena not ad-equately modeled by previously known mathematical structures.Turing's analysis of the notion of algorithm by means of Turingmachines is an obvious example. In this way, by reaching out andstudying new pheonemona, applied mathematics in general, andmathematical logic in particular, enriches mathematics, not onlywith new theorems, but also with new mathematical structures,structures for the mathematician to study and for others to applyin new domains.The theory of circular and otherwise extra-ordinary sets pre-sented in this book is an excellent example of this synergisticprocess. Aczel's work W88 motivated by work of Robin Milner incomputer science modeling concurrent processes. The fact thatthese processes are inherently circular makes them awkward tomodel in traditonal set theory, since most straightforward ideasrun afoul of the axiom of foundation. As a result, Milner's owntreatment was highly syntactic. Aczel's original aim was to finda version of set theory where these circular phenomena could bemodeled in a straightforward way, using standard techniques fromset theory. This forced him to develop an alternative conceptionof set, the conception that lies at the heart of this book. Aczelreturns to his starting point in the final chapter of this book.Before learning of Aczel's work, I had run up against similardifficulties in my work in situation theory and situation seman-tics. It seemed that in order to understand common knowledge(a crucial feature of communication), circular propositions, vari-ous aspects of perceptual knowledge and self-awareness, we hadto admit that there are situations that are not wellfounded underthe "constituent of' relation. This meant that the most naturalroute to modeling situations was blocked by the axiom of founda-tion. As a result, we either had to give up the tools of set theorywhich are so well loved in mathematical logic, or we had to enrichthe conception of set, finding one that admits of circular sets, atleast. I wrestled with this dilemma for well over a year beforeI argued for the latter move in (Barwise 1986). It was at justthis point that Aczel visited CSLI and gave the seminar whichformed the basis of this book. Since then, I have found severalapplications of Aczel's set theory, far removed from the problemsin computer science that originally motivated Aczel. Others havegone on to do interesting work of a strictly mathematical natureexploring this expanded universe of sets.I feel quite certain that there is still a lot to be done with thisuniverse of sets, on both fronts, that there are mathematical prob-lems to be solved, and further applications to be found. However,there is a serious linguistic obstacle to this work, arising out ofthe dominance of the cumulative conception of set. Just as thereused to be complaints about referring to complex numbers asnumbers, so there are objections to referring to non-well-foundedsets as sets. While there is clear historical justification for thisusage, the objection persists and distracts from the interest andimportance of the subject. However, I am convinced that readerswho approach this book unencumbered by this linguistic prob-lem will find themselves amply rewarded for their effort. TheAFA theory of non-well-founded sets is a beautiful one, full of po-tential for mathematics and its applications to symbolic systems.I am delighted to have played a small role, as Director of CSLIduring Aczel's stay, in helping to bring this book into existence.JON BARWISE