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).

This paper discusses a sequence of extensions of NFU , Jensen's improvement of Quine's set theory \New Foundations" (NF ) of [16]

This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirty-five years. The first three were “Set-theoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have

Sixty years ago in this journal, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was New Foundations for Mathematical Logic [6]. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid increase of interest in the alternatives to the hitherto customary Zermelo-Fr"ankel set theory, which promises a new lease of life for the axiomatic system now known as `NF'; its creator remains in good health too. Although he is best known to a wider public for his philosophical writings, his most enduring and most concrete legacy for the next fifty years may well turn out to be his most mathematical: he gave us NF. Set theory is the study of sets, which are the simplest of all mathematical entities. Let us illustrate by constrasting sets with groups. Two distinct groups can have the same elements and yet be told apart by the way those elements are related. Sets are distinguished from all other mathematical fauna by the fact that a set is constituted solely by its members: two sets with the same members are the same set. To use a bit of jargon from another age, sets are properties in extension. As a result, all set theories have the axiom of extensionality: (8xy)(x = y! (8z)(z 2 x! z 2 y)): they differ in their views on which properties have extensions. Since set theory first sprang on the scene about a hundred years ago there has been a tendency to attempt to use this simplicity to simplify and illuminate the rest of mathematics by translating (perhaps a better word is implementing) it into set theory. After all, if we can represent all of mathematics as facts about these delightfully simple things, some facts about mathematics might become clear that would otherwise remain obscure. This same simplicity means that set theory is always a good topic on which to try out any new mathematical idea.

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. 25--37. 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

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...

In what follows, ZF (L) is the natural extension of Zermelo-Fraenkel

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

sets and graph models of set and multiset theories

Abstract. A “new ” criterion for set existence is presented, namely, that a set {x | φ} should exist if the multigraph whose nodes are variables in φ and whose edges are occurrences of atomic formulas in φ is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so consistency of the set theory with weak extensionality and acyclic comprehension follows from the consistency of Jensen’s system NFU. It is much less obvious, but turns out to be the case, that this theory is equivalent to NFU: it appears at first blush that it ought to be weaker. This paper verifies that acyclic comprehension and stratified comprehension are equivalent, by verifying that each axiom in a finite axiomatization of stratified comprehension follows from acyclic comprehension.