The purpose of this paper is to present a discussion on the basic fundamentals of the theory of sets. Primarily, the discussion will be confined to the study of cardinal and ordinal numbers. The concepts of sets, classes of sets, and families of sets will be undefined quantities, and the concept of the class of all sets will be avoided.

The purpose of this composition is to develop a rigorous, axiomatic proof of the comparability of the cardinals of infinite sets.

This thesis is on ordered pairs of cardinal numbers.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2010. / Directed by Jerry Vaughan; submitted to the Dept. of Mathematics and Statistics. Title from PDF t.p. (viewed Jul. 14, 2010). Non-Latin script record Includes bibliographical references (p. 30).

The results from this dissertation are an exact computation of ultrapowers by measures on cardinals $\aleph\sb{n},\ n\in w$, in $L(\IR$), and a proof that ordinals in $L(\IR$) below $\delta\sbsp{5}{1}$ represented by descriptions and the identity function with respect to sequences of measures are cardinals. An introduction to the subject with the basic definitions and well known facts is presented in chapter I. In chapter II, we define a class of measures on the $\aleph\sb{n},\ n\in\omega$, in $L(\IR$) and derive a formula for an exact computation of the ultrapowers of cardinals by these measures. In chapter III, we give the definitions of descriptions and the lowering operator. Then we prove that ordinals represented by descriptions and the identity function are cardinals. This result combined with the fact that every cardinal $<\delta\sbsp{5}{1}$ in $L(\IR$) is represented by a description (J1), gives a characterization of cardinals in $L(\IR$) below $\delta\sbsp{5}{1}. Concrete examples of formal computations are shown in chapter IV.

Thesis (M.A.)--University of North Carolina Wilmington, 2008. / Title from PDF title page (viewed May 26, 2009) Includes bibliographical references (p. 64-72)

The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved.

Explains properties of mathematical sets, algebra of sets, and set order types.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES. / Bibliography: leaves 205-206. / by Fredric M. Katz. / Ph.D.

This thesis focuses on the logical design of binary adders. It covers topics extending from cardinal numbers to carry skip optimization. The conventional adder designs are described in detail, including: carry completion, ripple carry, carry select, carry skip, conditional sum, and carry lookahead. We show that the method of parallel prefix analysis can be used to unify the conventional adder designs under one parameterized model. The parallel prefix model also produces other useful configurations, and can be used with carry operator variations that are associative. Parallel prefix adder parameters include group sizes, tree shape, and device sizes. We also introduce a general algorithm for group size optimization. Code for this algorithm is available on the World Wide Web. Finally, the thesis shows the derivation for some carry operator variations including those originally given by Majerski and Ling. / text