A cumulative hierarchy of sets for constructive set theory

The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove new theorems in .     

Large sets in intuitionistic set theory

We consider properties of sets in an intuitionistic setting corresponding to large cardinals in classical set theory. Adding such ‘large set axioms’ to intuitionistic ZF set theory does not violate well-know metamathematical properties of intuitionistic systems. Moreover, we consider statements in constructive analysis equivalent to the consistency of such ‘large set axioms’.     

Canonical form of Tarski sets in Zermelo-Fraenkel set theory

We establish the equivalence of the notions of an inaccessible cumulative set and uncountable Tarski set. In addition, the equivalence of these notions and that of a galactic set is proved.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 323–333.Original Russian Text Copyright © 2005 by E. I. Bunina, V. K. Zakharov.This revised version was published online in April 2005 with a corrected issue number.     

Categorical characterization of quadrics

We give a characterization of smooth quadrics in terms of the existence of full exceptional collections of certain type, which generalizes a result of C.Vial for projective spaces.     

A lattice-valued set theory

A lattice-valued set theory is formulated by introducing the logical implication which represents the order relation on the lattice. Received: 27 September 1996 / Revised version: 14 July 1997     

Representations of concrete categories in the category of sets

An example of a concrete category which cannot be isomorphically embedded in the category of sets such that all monomorphisms are represented by injections is specified.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 125–127, July, 1969.     

A characterization of relatively compact sets of fuzzy sets

A purely topological characterization of relatively compact sets is given for the metric space (K(Y),D)$(\mathbb{K}(Y),D)$ of upper semicontinuous, compact-supported, normal fuzzy subsets of a metric space Y$Y$. The considered metric D$D$ is that of the distance between fuzzy subsets, which is the supremum of the Hausdorff distances of the corresponding level sets. In the given proof the compactness of a variational convergence which was introduced by De Giorgi and Franzoni is fundamental.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

Divergence of interpolation processes on sets of the second category

C([0, 1]) is the space of real continuous functions f(x) on [0, 1] and ω(δ) is a majorant of the modulus of continuity ω(f, δ), satisfying the condition \(\mathop {\overline {\lim } }\limits_{n \to \infty } \omega (1/n) \ln n = \infty \) . A solution is given to a problem of S. B. Stechkin: for any matrix \(\mathfrak{M}\) of interpolation points there exists an f(x) ? c([0, 1]), ω (f, δ) = o{ω(δ)} whose Lagrange interpolation process diverges on a set ? of second category on [0, 1].