Characterizations of commuting relations
Characterizing the polynomial hierarchy by alternating auxiliary pushdown automata
Characterizing the powerset by a complete (Scott) sentence
This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence if has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if is characterized by a Scott sentence, then is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁....
Characterizing tolerance trivial finite algebras
An algebra is tolerance trivial if where is the lattice of all tolerances on . If contains a Mal’cev function compatible with each , then is tolerance trivial. We investigate finite algebras satisfying also the converse statement.
Charakter von Mengensystemen.
Charakterisierung der Aufzählungsreduzierbarkeit.
Choice functions and well-orderings over the infinite binary tree
We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded...
Choice Mappings of Certain Classes of Finite Sets.
Choice principles in elementary topology and analysis
Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results — old and new — that specify how much “choice” is needed precisely to validate each of certain basic analytical and topological results.
Choice principles in Węglorz’ models
Węglorz' models are models for set theory without the axiom of choice. Each one is determined by an atomic Boolean algebra. Here the algebraic properties of the Boolean algebra are compared to the set theoretic properties of the model.
Choice sequences and Markov's principle
Choice sequences and reduction processes.
Choosing an attacker by a local derivation
Chromatic graphs, Ramsey numbers and the flexible atom conjecture.
Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC
In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. If in a partially ordered set, all chains are finite and all antichains have size , then the set has size for any regular . Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. Every partially ordered set...
Church-Rosser property and decidability of monadic theories of unary algebras
Circulant Boolean relation matrices
Circumcenter, Circumcircle and Centroid of a Triangle
We introduce, using the Mizar system [1], some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle. We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius...
Class preserving mappings of equivalence systems
By an equivalence system is meant a couple where is a non-void set and is an equivalence on . A mapping of an equivalence system into is called a class preserving mapping if for each . We will characterize class preserving mappings by means of permutability of with the equivalence induced by .
Classes de Wadge potentielles et théorèmes d'uniformisation partielle
On cherche à donner une construction aussi simple que possible d'un borélien donné d'un produit de deux espaces polonais. D'où l'introduction de la notion de classe de Wadge potentielle. On étudie notamment ce que signifie "ne pas être potentiellement fermé", en montrant des résultats de type Hurewicz. Ceci nous amène naturellement à des théorèmes d'uniformisation partielle, sur des parties "grosses", au sens du cardinal ou de la catégorie.