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Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
Linear Set Theory
, 1995
"... In this thesis, we develop four systems of set theory based on linear logic. All of those systems have the principle of unrestricted comprehension but they are shown to be consistent. The consitency proofs are given by establishing the cutelimination theorems. Our first system of linear set theory ..."
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In this thesis, we develop four systems of set theory based on linear logic. All of those systems have the principle of unrestricted comprehension but they are shown to be consistent. The consitency proofs are given by establishing the cutelimination theorems. Our first system of linear set theory SMALL is formulated in full linear logic, i.e., with exponentials. However we do not allow exponentials to appear inside of set terms. Secondly, we formulate a system of set theory in linear logic with infinitary additive conjunction and disjunction, instead of exponentials. This system is called AS 1 . Thirdly, we present the system of linear set theory LZF which is a conservative extension of ZermeloFraenkel set theory without the axiom of regularity or ZF \Gamma . The idea is to build up a linear set theory on top of ZF \Gamma in a style similar to SMALL. We establish a partial cutelimination result for LZF, and derive from it that LZF is a conservative extension of ZF \Gamma , ...
Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
Three Conceptual Problems That Bug Me
, 1996
"... Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought ..."
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Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought it would be worthwhile on this occasion to bring them to your attention side by side. In this talk I will explain the problems, together with some things that have been tried in the past and some new ideas for their solution. Types of conceptual problems. A conceptual problem is not one which is formulated in precise technical terms and which calls for a definite answer. For this reason, there are no clearcut criteria for their solution, but one can bring some criteria to bear. These will vary from case to case. There are three general types of conceptual problems in mathematics of which the ones that I will discuss today are examples. These are: 1 ffi<F
Handbook of the History of Logic. Volume 6
"... ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed p ..."
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ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semiinductive definitions, which require nontrivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of antifoundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of nonclassical logical systems, from contractionfree and manyvalued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of selfreferential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.