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Local Theory of Sets As a Foundation for Category Theory and Its Connection with the Zermelo–Fraenkel Set Theory

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Local Theory of Sets As a Foundation for Category Theory and Its Connection with the Zermelo–Fraenkel Set Theory Journal of Mathematical Sciences, Vol. 138, No. 4, 2006 LOCAL THEORY OF SETS AS A FOUNDATION FOR CATEGORY THEORY AND ITS CONNECTION WITH THE ZERMELO–FRAENKEL SET THEORY V. K. Zakharov, E. I. Bunina, A. V. Mikhalev, and P. V. Andreev UDC 510.223, 512.581 CONTENTS Introduction..............................................5764 1. First-Order Theories and LTS ....................................5767 1.1. Language of first-order theories ................................5767 1.2. Deducibility in a first-order theory . ............................5768 1.3.Aninterpretationofafirst-ordertheoryinasettheory...................5770 2.LocalTheoryofSets.........................................5772 2.1.Properaxiomsandaxiomschemesofthelocaltheoryofsets................5772 2.2.Someconstructionsinthelocaltheoryofsets........................5775 3. Formalization of the Naive Category Theory within the Framework oftheLocalTheoryofSets.....................................5776 3.1.Definitionofalocalcategoryinthelocaltheoryofsets...................5776 3.2. Functors and natural transformations and the “category of categories” and the “category of functors” in the local theory of sets generated by them . ................5777 4. Zermelo–Fraenkel Set Theory and Mirimanov–von Neumann Sets ................5779 4.1.TheproperaxiomsandaxiomschemesoftheZFsettheory................5779 4.2.OrdinalsandcardinalsintheZFsettheory.........................5782 4.3.Mirimanov–vonNeumannsetsintheZFsettheory.....................5786 5. Universes, Ordinals, Cardinals, and Mirimanov–von Neumann Classes intheLocalTheoryofSets.....................................5791 5.1. The relativization of formulas of the LTS to universal classes. The interpretation of the ZF set theory in universal classes . ............................5791 5.2. The globalization of local constructions ............................5793 5.3.Ordinalsandcardinalsinthelocaltheoryofsets......................5795 5.4. Mirimanov–von Neumann classes in the local theory of sets and their connection with universal classes . ........................................5798 5.5. The structure of the assemblies of all universal classes and all inaccessible cardinals in the localtheoryofsets.......................................5801 6.RelativeConsistencybetweenLTSandtheZFTheory......................5803 6.1.AdditionalaxiomsoninaccessiblecardinalsintheZFtheory................5803 6.2. “Forks” of relative consistency . ................................5807 7.ProofofRelativeConsistencybytheMethodofAbstractInterpretation............5816 7.1.Abstractsofasettheory....................................5816 7.2.Theabstractinterpretationofafirst-ordertheoryinasettheory.............5817 8. Undeducibility of Some Axioms in LTS . ............................5818 8.1. The undeducibility of the axiom scheme of replacement . ................5818 Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 119, Set Theory, 2004. 1072–3374/06/1384–5763 c 2006 Springer Science+Business Media, Inc. 5763 8.2.ThenondependenceofaxiomICUontheaxiomsoftheLTS................5826 8.3. Locally-minimal theory of sets . ................................5827 References . ............................................5828 Introduction The crises that arose in naive set theory at the beginning of the 20th century brought to the fore some strict axiomatic constructions of theories of mathematical totalities. The most widely used of them are the theory of sets in Zermelo–Fraenkel’s axiomatics (ZF) (see [23]) and the theory of classes and sets in von Neumann–Bernays–G¨odel’s axiomatics (NBG) (see [20] and [28]). These axiomatic theories eliminated all the known paradoxes of naive set theory at the expense of a sharp restriction on possible expressive means. At the same time, they provided an opportunity to include almost all the then existing mathematical objects and constructions within the framework of these theories. In 1945, a new mathematical notion of category was introduced by Eilenberg and MacLane in the initial paper [11]. From that time, category theory became an independent field of mathematics. But from the very beginning of its origin, category theory encountered the unpleasant circumstance that it did not fall not only within the framework of the theory of sets in Zermelo–Fraenkel’s axiomatics but even within the framework of the theory of classes and sets in von Neumann–Bernays–G¨odel’s axiomatics (NBG) (see [11]). For this reason, in 1959, in the paper [25], MacLane pose the general problem of constructing a new and more flexible axiomatic set theory that could serve as an adequate logical foundation for the whole naive category theory. Different variants of new axiomatic theories of mathematical totalities, adjusted for one or another need of category theory, were proposed by Ehresmann [10], Dedecker [9], Sonner [29], Grothendieck [14], da Costa ([6] and [7]), Isbell [18], MacLane ([26] and [27]), Feferman [12], Herrlich and Strecker [17], and others. Ehresmann, Dedecker, Sonner, and Grothendieck introduced an important notion of (categorical) uni- verse U, i. e., a totality of objects which satisfies the following properties of closedness: (1) X ∈ U ⇒ X ⊂ U (Ehresmann–Dedecker did not propose this property); (2) X ∈ U ⇒P(X), ∪X ∈ U; (3) X, Y ∈ U ⇒ X ∪ Y,{X, Y }, X, Y ,X × Y ∈ U; (4) X ∈ U ∧ (F ∈ U X ) ⇒ rng F ∈ U; (5) ω ∈ U (ω = {0, 1, 2,...} is here the set of all finite ordinal numbers). Within the framework of such a universe, it is possible to develop a quite rich category theory, in particular, the theory of categories with direct and inverse limits. To satisfy all needs of category theory, these authors proposed to strengthen the ZF or NBG set theories by the strong universality axiom pos- tulating that every set belongs to some universal set. In MacLane’s axiomatics, the existence of at least one universal set is postulated. Similar versions were proposed by Isbell and Feferman. Herlich–Srecker’s axiomatics dealt with objects of three types: sets, classes, and conglomerations. Within the framework of each of these axiomatics, some definitions of category and functor are given. But the notions of category given in [12, 17, 18, 26, 27] are not closed with respect to such important operations of naive category theory as the “category of categories” and the “category of functors” (see [16], 8.4). Within the framework of the axiomatic theories [9, 10, 14, 29] the definition of U-category, U-functor, and natural U-tranformation consisting of subsets of a universal set U is given. This notion of category is closed with respect to such operations as the V-category of U-categories and the V-category of U-functors, where V is some universal set containing the universal set U as an element. 5764 The axiomatics from [6] and [7] also give definitions of a category and a functor closed under the mentioned operations of naive category theory. But da Costa uses the logic with the nonconstructive rule of deduction ϕ(V1), ϕ(V2), ϕ(V3), ··· ∀tϕ(t), where V1,V2,...,Vn,... is an infinite sequence of constants, which denotes universes like the NBG-universe (see [7]). In connection with the logical difficulties of conctructing a set-theoretical foundation for the whole naive category theory, different attempts to construct purely arrow-axiomatic foundations were undertaken (see Lawvere [24], Blanc and Preller [2], and Blanc and Donadieux [3]). But in attempts at an arrow- axiomatic description of indexed categories and fiber categories, their own logical difficulties appeared (see [16], 8.4). Now let us consider more precisely what the formulation of MacLane’s problem mentioned above means. To do this, we need to have a strict definition of category. This definition became possible after the elementary (≡ first-order) category theories Tc took shape. There are many such theories, two-sorted and one-sorted (see, for example, [27, I.3, I.8], [16, 8.2], [30, 2.3, 11.1]). But for every set theory S, there exist canonical one-to-one correspondences between the totalities of models of these theories in the theory S. Therefore, for the strict definition of a category one can use any elementary theory Tc. According to the definition of Lawvere [24] (see also [16, 8.2]), a category in some set theory S is any model of the theory Tc in the S theory (see 1.3 below). The complete (partial) formalization of naive category theory C in the set theory S is an adequate translation of all (some) notions and constructions of naive category theory C into strict notions and constructions for categories (as models of the theory Tc) in the set theory S. Along with the notion of a category in the set theory S, there also exists the notion of abstract category in S.Anabstract category in the set theory S is any abstract model of the theory Tc in the set theory S (see 7.1. below). The complete (partial) abstract formalization of naive category theory C in the set theory S is an adequate translation of all (some) notions and constructions of naive category theory C into strict notions and constructions for abstract categories (as abstract models of the theory Tc) in the S set theory. According to these definitions abstract categories in the ZF set theory can be considered also on classes (as abstracts of the ZF theory), and abstract categories in the NBG set theory can be considered also on assemblies (as abstracts of the NBG theory). However, the complete abstract formalization of naive category theory C in any S set theory is impossible, because
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