Journal of Mathematical Sciences, Vol. 138, No. 4, 2006 CHARACTERIZATION OF MODEL MIRIMANOV–VON NEUMANN CUMULATIVE SETS E. I. Bunina and V. K. Zakharov UDC 510.223 CONTENTS Introduction..............................................5831 1. Some Facts From Zermelo–Fraenkel Set Theory . ........................5832 1.1. Classes in the ZF set theory . ................................5832 1.2.OrdinalandcardinalsintheZFsettheory..........................5835 2.CumulativeSetsandTheirProperties...............................5837 2.1.Constructionofcumulativesets................................5837 2.2.Propertiesofcumulativesets..................................5837 2.3.Propertiesofinaccessiblecumulativesets...........................5841 3.UniversalSetsandTheirConnectionwithInaccessibleCumulativeSets............5843 3.1.Universalsetsandtheirproperties...............................5843 3.2.Descriptionoftheclassofalluniversalsets..........................5845 3.3. Enumeration of the class of all universal sets in the ZF+AU theory and the structural formoftheuniversalityaxiom.................................5846 3.4. Enumeration of the class of all inaccessible cardinals in the ZF+AI theory and the struc- tural form of the inaccessibility axiom . ............................5849 4. Weak Forms of the Axioms of Universality and Inaccessibility . ................5851 4.1. Axioms of ω-universality and ω-inaccessibility ........................5851 4.2. Comparison of various forms of the universality and inaccessibility axioms ........5854 5. Description of the Class of all Supertransitive Standard Models oftheNBGTheoryintheZFTheory...............................5858 5.1. Supertransitive standard models of the ZF theory having the strong subtitution property 5858 5.2.SupertransitivestandardmodelsoftheNBGtheoryintheZFtheory...........5863 6. Tarski Sets and Galactic Sets. Theorem on the Characterization ofNaturalModelsoftheNBGTheory...............................5870 6.1.Tarskisetsandtheirproperties................................5870 6.2.GalacticsetsandtheirconnectionwithTarskisets.....................5873 6.3. Characterization of Tarski sets. Characterization of natural models of the NBG theory . 5875 7. Characterization of Natural Models of the ZF Theory . ....................5876 7.1.Scheme-inaccessiblecardinalnumbersandscheme-inaccessiblecumulativesets......5876 7.2. Scheme-universal sets and their connection with scheme-inaccessible cumulative sets . 5880 7.3.SupertransitivestandardmodelsoftheZFtheoryintheZFtheory............5884 7.4. Tarski scheme sets. Theorem on the characterization of natural model of the ZF theory . 5888 References . ............................................5890 Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 119, Set Theory, 2004. 5830 1072–3374/06/1384–5830 c 2006 Springer Science+Business Media, Inc. Introduction The crisis that appeared in the naive set theory at the beginning of the 20th century has led to the construction of several strict axiomatic set theories. The most useful among them is the set theory in the Zermelo–Fraenkel axiomatics (ZF) (1908, 1922 [8]) and the class and set theory in the von Neumann– Burnice–Hodel¨ axiomatics (NBG) (1928, 1937, 1940 [10]). In 1917, using transfinite induction Mirimanov [20] constructed the cumulative collection (≡hierarchy) of sets Vα for all order numbers α having the following properties: (1) V0 ≡∅; (2) Vα+1 = Vα ∪P(Vα)(P(Vα) denotes the set of all subsets of the set Vα); (3) Vα = ∪ Vβ|β ∈ α for any limit order number α. It turns out that cumulative sets Vα themselves and the collection Vα|α ∈ On as a whole have many remarkable properties. In particular, von Neumann proved in [22] that the regularity axiom in ZF is equivalent to the property ∀x∃α(α is an ordinal number ∧x ∈ Vα) and the class ∪ Vα|α ∈ On is an abstract (≡ class) standard model for the ZF theory in ZF. Models of the ZF and NBG theories of the form (Vα, =, ∈) are said to be natural. After the introduction of the concept of a (strongly) inaccessible cardinal number by Zermelo in [28] and by Sierpinski–Tarski in [24], Zermelo [28] (not strictly) and Shepherdson [23] (strictly) proved that a set U is a supertransitive standard model for the NBG theory iff it has the form Vκ+1 for a certain inaccessible cardinal number κ. Thus, the natural model of the NBG theory was explained. The Zermelo–Shepherdson theorem admits the following equivalent reformulation: a set U is a super- transitive standard model for the ZF theory with the strong replacement property (∀X∀f(X ∈ U ∧ f ∈ U X ⇒ rng f ∈ U)) iff it has the form Vκ for a certain inaccessible cardinal number κ. Starting from the requirements of category theory, instead of the metaconcept of a supertransitive standard model set with the strong replacement property for the ZF theory, Ehresmann [6], Dedecker [5], Sonner [25], and Grothendieck (see [9]), introduced an equivalent set-theoretic concept of a universal set U (see [18], I.6, [7], [12]), which is defined by the following properties: (1) x ∈ U ⇒ x ⊂ U; (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 (strong replacement property); (5) ω ∈ U (here, ω ≡{0, 1, 2,...} is the set of all finite ordinal numbers). To deal with categories in the set-theoretic framework, they suggested to strengthen the ZF theory by adding the universality axiom AU, according to which each set is an element of a certain universal set. The equivalent form of the Zermelo–Shepherdson theorem states that the universality axiom AU is equivalent to the inaccessibility axiom AI according to which for every ordinal number, there exists an inaccessible cardinal number strictly greater than it. For axiomatic construction of inaccessible cardinal numbers, in [26] (see also [17], IX, § 1and§ 5), Tarski introduced the concept of a Tarski set U, which is defined by the following properties: (1) x ∈ U ⇒ x ⊂ U (transitivity property); (2) x ∈ U ⇒P(x) ∈ U (exponentiality property); (2) (x ⊂ U ∧∀f(f ∈ U x ⇒ rng f = U) ⇒ x ∈ U (Tarski property). In [26], Tarskii proved that the set Vκ (≡ inaccessible cumulative set) is a Tarski set for each cardinal number κ. Also, in [26], Tarski proved that the inaccessibility axiom AI is equivalent to the Tarski axiom AT, according to which every set is an element of a certain Tarskii set. In connection with the Tarski theorem, the following problem remains open unit now: to what extent is the axiomatic concept of Tarski set is wider than the constructive concept of inaccessible cumulative set? 5831 In this paper, we give an answer to this question: the concepts of an inaccessible cumulative set and that of an uncountable Tarski set are equivalent. The equivalence of the concepts of an inaccessible cumulative set and an uncountable Tarski set is proved by using the concept of a universal set. More precisely, it is proved that every uncountable Tarski set is universal. As a result, we obtain the following theorem on the characterization of natural models of the NBG theory: the following properties are equivalent for a set U: (1) U is an inaccessible cumulative set, i.e., U = Vκ for a certain inaccessible cardinal number κ; (2) P(U) is a supertransitive standard model for the NBG theory; (3) U is a supertransitive standard-model with the strong replacement property of the ZF theory; (4) U is a universal set; (5) U is an uncountable Tarski set. The Zermelo–Shepherdson theorem yields a canonical form of supertransitive standard models of the NBG theory and an (equivalent) canonical form of standard models with the strong replacement property of the ZF theory. However, Montague and Vaught proved in [21] that for any inaccessible cardinal number κ, there exists an ordinal number θ<κ such that it is inaccessible and the cumulative set Vθ is a supertransitive standard model of the ZF theory. Therefore, the problem on the canonical forms of supertransitive standard models of the ZF theory turned out to be more complicated. Since the concept of model in the ZF theory cannot be defined by a finite set of formulas, in this paper, using the formula schema and its relativization to the set Vθ, we introduce the concept of a (strongly) scheme-inaccessible cardinal number θ and prove a scheme analog of the Zermelo–Shepherdson theorem. To prove this theorem, we introduce the concept of a scheme-universal set, which is a scheme analog of the concept of a universal set. Moreover, in this paper, we introduce the concept of a scheme Tarski set, which is a scheme analog of the concept of a Tarski set. As a result, we prove the theorem on the characterization of natural models of the ZF theory: the following properties are equivalent for a set U: (1) U is a scheme-inaccessible cumulative set, i.e., U = Vθ for a certain scheme-inaccessible cardinal number θ; (2) U is a supertransitively standard model for the ZF theory; (3) U is a scheme-universal set; (4) U is a scheme Tarski set. In this paper, the problems mentioned above are solved for the ZF set theory (with the axiom of choice). For the NBG set theory, all things are equally true. For the reader’s convenience, we present all the necessary facts that are not sufficiently reflected in the literature or merely refer to the mathematical folklore, with complete proofs. 1. Some Facts From Zermelo–Fraenkel Set Theory 1.1. Classes in the ZF set theory. We first present a list of proper axioms and axiom schemes of the ZF theory, the Zermelo–Fraenkel theory with the axiom