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Journal article
Authors Paul Kindvall Gorbow
Published in Journal of Symbolic Logic
Volume 84
Issue 2
Pages 798-832
ISSN 0022-4812
Publication year 2019
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 798-832
Language en
Keywords category theory, set theory, new foundations, NF, NFU, INF, INFU, algebraic set theory, topos, models, Mathematics, Science & Technology - Other Topics, RSTER T., 2014, The category of sets in stratifiable set theories, RSTER T., 1995, Set Theory with a Universal Set, larty c, 1992, journal of symbolic logic, v57, p555
Subject categories Languages and Literature


This paper consists in the formulation of a novel categorical set theory, MLCat, which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic (denoted with an "I" on the left) or classical NF, with atoms (denoted with a "U" on the right) or not: NF is a set theory that rescues the intuition behind naive set theory, by imposing a so called stratification constraint on the formulae featuring in the comprehension schema. It turns out that NFU is quite a different theory from NF; for example, NFU is consistent with the axiom of choice, while NF is not. Not very much is known about INF and INFU. The axioms of the categorical theory developed here express that its structures have an endofunctor, with certain coherence properties. By means of this endofunctor, an appropriate axiom of power objects is formulated for the setting of NF. The most interesting direction of the equiconsistency result is established by interpreting the set theory in the category theory, through the machinery of categorical semantics, thus making essential use of the flexibility inherent in category theory. An example of this flexibility is that we obtain a transparent proof that (I)NF is equiconsistent with (I)NFU + vertical bar V vertical bar = vertical bar PV vertical bar.(1) Moreover, it is shown that (I)ML(U)(Cat) is connected to topos theory as follows: THEOREM. For any category C vertical bar= (I)ML(U)(Cat), with endofunctor T, the full subcategory on the fixed-points(2) of T is a topos. The relevance of this categorical work lies in that it provides a basis for studying the dynamics of NF within the realm of category theory. In particular, it opens up for constructions of categorical models of intuitionistic versions of NF, and for stratified approaches to type-theory. It may also be relevant for attempts to prove or simplify proofs of the consistency of classical NF.

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