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The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
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SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
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? ..."
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The Mark2 Theorem Prover
"... ABSTRACT1 @ ?x: ?x = 68 Id @ ?x ["ID"] *)  s "?x";  rri "ID";ex();  p "ABSTRACT1@?x"; (* ABSTRACT2 @ ?x: ?f @ ?a = COMP != ?f @ (ABSTRACT @ ?x) =? ?a ["COMP"] *)  s "?f@?a";  rri "COMP";  right(); right(); r ..."
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ABSTRACT1 @ ?x: ?x = 68 Id @ ?x ["ID"] *)  s "?x";  rri "ID";ex();  p "ABSTRACT1@?x"; (* ABSTRACT2 @ ?x: ?f @ ?a = COMP != ?f @ (ABSTRACT @ ?x) =? ?a ["COMP"] *)  s "?f@?a";  rri "COMP";  right(); right(); ri "ABSTRACT@?x";  prove "ABSTRACT2@?x"; (* ABSTRACT3 @ ?x: ?a & ?b = RAISE0 =? ((ABSTRACT @ ?x) =? ?a) & (ABSTRACT @ ?x) =? ?b [] *)  s "?a&?b";  right();  ri "ABSTRACT@?x";  up();left();  ri "ABSTRACT@?x";  top();  ri "RAISE0";  prove "ABSTRACT3@?x"; (* ABSTRACT4 @ ?x: ?a = [?a] @ ?x [] *)  s "?a";  ri "BIND@?x"; ex(); 69  p "ABSTRACT4@?x"; (* ABSTRACT@term will (attempt to) express a target term as a function of its parameter "term" *) (* ABSTRACT @ ?x: ?a = (ABSTRACT4 @ ?x) =?? (ABSTRACT3 @ ?x) =?? (ABSTRACT2 @ ?x) =?? (ABSTRACT1 @ ?x) =? ?a ["COMP","ID"] *)  s "?a";  ri "ABSTRACT1@?x";  ari "ABSTRACT2@?x";  ari "ABSTRACT3@?x";  ari "ABSTRACT4@?x";  p "ABSTRACT@?x"; (* REDUCE will reverse the effect of ABSTRACT; it will "evaluate" functions built by ABSTRACT *) (* REDUCE: ?f @ ?x = (ABSTRACT4 @ ?x) !!= ((RL @ REDUCE) *? RAISE0) !!= ((RIGHT @ REDUCE) *? COMP) =?? ID =? ?f @ ?x ["COMP","ID"] *)  dpt "REDUCE";  s "?f@?x";  ri "ID";  ari "(RIGHT@REDUCE)*?COMP";  arri "(RL@REDUCE)*?RAISE0";  arri "ABSTRACT4@?x";  prove "REDUCE"; (* old approach to hypotheses *) (* equational forms of tactics given without proof; the proofs of the tactics involve no actual rewriting *) PIVOT: (?a = ?b)  ?T , ?U = (RIGHT @ LEFT @ EVAL) =? HYP =? (?a = ?b) 70  ((BIND @ ?a) =? ?T) , ?U ["HYP"] REVPIVOT: (?a = ?b)  ?T , ?U = (RIGHT @ LEFT @ EVAL) =? HYP != (?a = ?b)  ((BIND @ ?b) =? ?T) , ?U ["HYP"] We now present examples of the use of thes...
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
About the coexistence of classical sets with nonclassical ones: a survey’, Logic and Logical Philosophy, this issue
 Logic and Logical Philosophy 11 (2003
"... Abstract. This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory ” (the universes discussed here concern, roughly speaking: stratified sets, partial sets, positive sets, paradoxical sets and double sets). ..."
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Abstract. This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory ” (the universes discussed here concern, roughly speaking: stratified sets, partial sets, positive sets, paradoxical sets and double sets).
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
Russell's Paradox Of The Totality Of Propositions
, 2000
"... on this analysis, and although Oksanen quoted Russell's description of the paradox in detail, he did not show how it is explained in NFU after his resolution of the other related modal paradoxes; in fact, it is not at all clear how this might be done in NFU. 1PoM, p. 527. 2See, e.g., Grim ..."
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on this analysis, and although Oksanen quoted Russell's description of the paradox in detail, he did not show how it is explained in NFU after his resolution of the other related modal paradoxes; in fact, it is not at all clear how this might be done in NFU. 1PoM, p. 527. 2See, e.g., Grim 1991, pp. 92f. 3See, e.g., Grim 1991, p. 119 and Jubien 1988, p. 307. 4See Oksanen 1999. NFU is a modified version of Quine's system NF. It was first described in Jenson 1968 and recently has been extensively developed in Holmes 1999. Nordic Journal of Philosophical Logic, Vol. 5, No. 1, pp. 2537. 2000 Taylor & Francis. 26 nino b. cocchiarella One reason why Russell's argument is dicult to reconstruct in NFU is that it is based on the logic of propositions, and implicitly in that regard on a theory of predication rather than a theory of membership. A mo
FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE
"... Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unli ..."
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Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited ” or “naive ” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. From the very beginnings of the subject of category theory as introduced by Eilenberg & Mac Lane (1945) it was recognized that the notion of category lends itself naturally to
The usual model construction for NFU preserves information
, 2009
"... The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements ..."
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The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements (it appears that information about their extensions is discarded). The surprising result of this paper is that this information is not discarded at all: the membership relation of the original model (restricted to the domain of the model of NFU) is definable in the language of NFU. A corollary of this is that the urelements of a model of NFU obtained by the “usual ” construction are inhomogeneous: this was the question the author was investigating initially. Other aspects of the mutual interpretability of NFU and a fragment of ZFC are discussed in sufficient detail to place
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
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By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4