Set-mappings on Dedekind sets

Fairly deep results of Zermelo-Fraenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K=K, where K is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are largely faithful in style to the original mathematics.

Fuzzy Information and Engineering

In this paper, based upon Fs-set theory [Yogesara V, Srinivas G, Rath B. A theory of Fs-sets, Fs-complements and Fs-de Morgan laws. IJARCS. 2013;4(10)], we define Fs-Cartesian product of given family Fs-subsets of give Fs-set and we prove Axiom of choice for Fs-sets and we study the validity of converse of the Axiom of choice for Fs-sets.

Notre Dame Journal of Formal Logic, 1988

We make the following definition: Definition Let (Q,<) be a quasi-order (i.e., < is reflexive and transitive). Two elements x,y of Q are said to be incompatible if there does not exist z E Q such that z < x and z < y. A subset / of Q is said to be an incompatible set if any two elements of / are incompatible. For each x E Q, let l(x) denote the set of lower bounds of x; and let c(x) denote the set of elements of Q that are compatible with x. "Countable" is used here to mean "countably infinite".

Proceedings of the American Mathematical Society, 1989

Let M M be a countable transitive model of ZFC and A A be a countable M M -generic family of Cohen reals. We prove that there is no smallest transitive model N N of ZFC that either M ∪ A ⊆ N M \cup A \subseteq N or M ∪ { A } ⊆ N M \cup \{ A\} \subseteq N . It is also proved that there is no smallest transitive model N N of ZFC − ^{-} (ZFC theory without the power set axiom) such that M ∪ { A } ⊆ N M \cup \{ A\} \subseteq N . It is also proved that certain classes of extensions of M M obtained by Cohen generic reals have no minimal model.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992

We prove the independence of some weakenings of the axiom of choice related to the question if the unions of wellorderable families of wellordered sets are wellorderable.

The Journal of Symbolic Logic, 1985

The purpose of this paper is not to produce a survey of systems with a universal set: we do not yet understand them well enough. Rather it will concentrate on one particular aspect of them: the curious circumstance that there are half-a-dozen or so distinct proofs of ~ AC available in set theories with a universal set. This began to emerge in 1953 when Specker published in [2] a proof of ~ AC in Quine&#39;s system NF. Until recently this was an isolated phenomenon and poorly understood. The proof answered few questions and seemed rather ad hoc, thus inviting an investigation to determine whether this was an artefact caused by the particular axioms for NF, or part of a general conflict between the demands of big sets and AC. We will start with an informal discussion of the genesis of set theories with a universal set, collecting, en route, a number of desiderata for such theories. A number of new refutations of AC in systems meeting some or all of these conditions will then be presen...

Set theory deals with the most fundamental existence questions in mathematicsquestions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called "arbitrary sets." This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what is meant by definability and by "arbitrariness," a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo-Fraenkel system goes in laying out principles that capture the idea of "arbitrary sets". We argue that the theory is rather poor in this respect.

Mathematical Logic Quarterly, 1993

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results.

Journal of Symbolic Logic, 2000

It is consistent that there is a set mapping from the four-tuples of ωninto the finite subsets with no free subsets of sizetnfor some natural numbertn. For anyn&lt; ω it is consistent that there is a set mapping from the pairs of ωninto the finite subsets with no infinite free sets. For anyn&lt; ω it is consistent that there is a set mapping from the pairs of ωninto ωnwith no uncountable free sets.

2008

In this paper we characterize the existence of best choices of arbitrary binary relations over non finite sets of alternatives, according to the Generalized Optimal-Choice Axiom condition introduced by Schwartz. We focus not just in the best choices of a single set X, but rather in the best choices of all the members of a family K of subsets of X. Finally we generalize earlier known results concerning the existence (or the characterization) of maximal elements of binary relations on compact subsets of a given space of alternatives.