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## On Ordered Pairs of Cardinal Numbers

Description: This thesis is on ordered pairs of cardinal numbers.
Date: 1957
Creator: Dickinson, John Dean
Partner: UNT Libraries

## A Fundamental Study of Cardinal and Ordinal Numbers

Description: 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.
Date: August 1966
Creator: Thornton, Robert Leslie
Partner: UNT Libraries

## The Comparability of Cardinals

Description: The purpose of this composition is to develop a rigorous, axiomatic proof of the comparability of the cardinals of infinite sets.
Date: May 1964
Creator: Owen, Aubrey P.
Partner: UNT Libraries

## Descriptions and Computation of Ultrapowers in L(R)

Description: 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.
Date: August 1995
Creator: Khafizov, Farid T.
Partner: UNT Libraries

## Equivalent Sets and Cardinal Numbers

Description: 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.
Date: December 1975
Creator: Hsueh, Shawing
Partner: UNT Libraries

## Some Properties of Transfinite Cardinal and Ordinal Numbers

Description: Explains properties of mathematical sets, algebra of sets, and set order types.
Date: 1940
Creator: Cunningham, James S.
Partner: UNT Libraries