Displaying similar documents to “On rigid relation principles in set theory without the axiom of choice”

Definitions of finiteness based on order properties

Omar De la Cruz, Damir D. Dzhafarov, Eric J. Hall (2006)

Fundamenta Mathematicae

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A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them...

Binary and ternary relations

Vítězslav Novák, Miroslav Novotný (1992)

Mathematica Bohemica

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Two operators are constructed which make it possible to transform ternary relations into binary relations defined on binary relations and vice versa. A possible graphical representation of ternary relations is described.

Basic Operations on Preordered Coherent Spaces

Klaus Grue, Artur Korniłowicz (2007)

Formalized Mathematics

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This Mizar paper presents the definition of a "Preordered Coherent Space" (PCS). Furthermore, the paper defines a number of operations on PCS's and states and proves a number of elementary lemmas about these operations. PCS's have many useful properties which could qualify them for mathematical study in their own right. PCS's were invented, however, to construct Scott domains, to solve domain equations, and to construct models of various versions of lambda calculus.For more on PCS's,...

On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Horst Herrlich, Paul Howard, Eleftherios Tachtsis (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice ( AC ), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.

Finiteness and choice

Omar De la Cruz (2002)

Fundamenta Mathematicae

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We deal with weak choice principles of the form: Every "finite" family of non-empty sets has a choice function, where "finite" stands for one of several different definitions of finiteness that are not equivalent unless we assume the axiom of choice (AC). Several relations of implication and independence are established. In the process, we answer a few open questions about the relations between different definitions of finiteness.