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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
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
4, SOME FORMAL SYSTEMS FOR THE UNLIMITED THEORY OF STRUCTURES AND CATEGORIES
"... Abstract. In the informal unlimited theory of structures and (particularly) categories, one considers unrestricted statements concerning structures such as that the substructure relation on all structures of a given kind forms a partially ordered structure. or that the collection of all categories f ..."
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Abstract. In the informal unlimited theory of structures and (particularly) categories, one considers unrestricted statements concerning structures such as that the substructure relation on all structures of a given kind forms a partially ordered structure. or that the collection of all categories forms a category with arbitrary These sorts of propositio~s are not accounted for difunctors as its morphisms. The aim of the present work is to give a founrectly by currently accepted means. dation for the theory of structures including such unlimited statementsmore or less as they are presented to usby means of certain formal systems. The theories studied here are based on an extension of Quinels idea of stratification. Their use is justified bya consistency proof. adapting methods of Jensen. These systems are successful for the basic aim to a considerable extent. but they suffer a specific defect which prevents them from being fully successful. Some possible alternatives are also suggested.
On Multidominance and Linearization Mark
"... This article centers around two questions: What is the relation between movement and structure sharing, and how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Without ad hoc restrictio ..."
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This article centers around two questions: What is the relation between movement and structure sharing, and how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Without ad hoc restrictions on the input, both options follow from Merge. They can be represented in terms of multidominance. Although more structural freedom ensues than standardly thought, the grammar is not completely unconstrained: Arguably, proliferation of roots is prohibited. Furthermore, it is explained why external remerge has somewhat different consequences than internal remerge. For instance, apparent nonlocal behavior is attested. At the PF interface, the linearization of structures involving remerge is nontrivial. A central problem is identified, apart from the general issue why remerged material is only pronounced once: There are seemingly contradictory linearization demands for internal and external remerge. This can be resolved by taking into account the different structural configurations. It is argued that the linearization is a PF procedure involving a recursive structure scanning algorithm that makes use of the inherent asymmetry between sister nodes imposed by the operation of Merge. Keywords: linearization; movement; multidominance; PF interface; (re) merge
semantic
, 2012
"... Symmetry motivates a new consistent fragment of NF and an extension of NF with ..."
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Symmetry motivates a new consistent fragment of NF and an extension of NF with