Alfred Tarski: Cardinal Algebras
This work has its origin in certain studies in general set theory and, more specifically, in the arithmetic of cardinal numbers. The present work was conceived about twenty years ago, but circumstances beyond the control of the author have rendered its earlier realization impossible. The publications of the author related to the central ideas of this work are listed in the bibliography. The understanding of this book does not require an extensive knowledge of set theory, but a certain orientation in fundamental notions and methods of this discipline is desirable and will prove helpful in reading the last three sections of the work. For general information in this domain the reader may consult the first few chapters of the books by Hausdorff, Schoenflies, or Sierpinski. Schoenflies' work contains many valuable historical references, while Sierpinski's book will throw light on the discussions involving the axiom of choice.
Among the results which have been obtained so far in the arithmetic of cardinals, two main types can be distinguished. On the one hand, we have a series of very strong and general theorem.,-, which exhaust large portions of the arithmetic of cardinals, e.g., the theory of cardinal addition; these theorems have been established by .applying the so-called axiom of choice in its most general form and, in particular, the well-ordering principle. As examples we may mention the theorems stating that the cardinals are well ordered and that the sum of any two infinite cardinals equals the larger of them. On the other hand, results are known which are of a more restricted (though by no means more trivial) character, but which have been obtained in a more constructive way, mostly without the help of the axiom of choice; the method involved consists, roughly speaking, in considering directly transformations which establish one-to-one correspondences between given sets and in constructing in terms of these sets and transformations new sets and new transformations. The best-known example is the Cantor-Bernstein equivalence theorem by which any two cardinals are equal if each of them is at most equal to the other.
The results of the second type are interesting not only from the point of view of foundations. By analyzing their proofs we usually arrive at more general formulations which belong to the general theory of one-to-one transformations, and which have found some interesting applications outside the domain of abstract set theory - for instance, in the theory of measure.
All the results of the second type which concern cardinal addition prove to be derivable in a purely arithmetical way - from a small number of basic theorems. The derivations are not simple but, in general, are not more involved than direct proofs carried through by the method indicated above. Also some new and interesting results can be obtained in this way. The proof of the basic theorems themselves presents no difficulties. The basic theorems have the character of formal laws which apply not only to the arithmetic of cardinals but also to various other mathematical systems. All this suggests the idea of introducing and studying a new kind of abstract algebras for which these basic theorems would serve as defining postulates. A realization of this idea is the main purpose of the present work.
The algebras in question will be referred to as Cardinal Algebras. Each of them is constituted by a set of arbitrary elements and by two operations, that of binary addition and that of addition of infinite sequences. (Each of these operations can be defined in terms of the other.) The algebras are assumed to satisfy a number of familiar laws of an elementary nature - closure postulates, commutative and associative laws, and the postulate of the zero element; and, in addition, two existential postulates which are characteristic of cardinal algebras - the refinement postulate and the remainder, or infinite chain, postulate.
The variety of arithmetical laws which follow from these postulates is very large; and so is the variety of mathematical systems in which these postulates are satisfied. As elementary examples of cardinal algebras we list non-negative integers and non-negative real numbers (under ordinary addition) with 8 included in both cases; non-negative real functions over an arbitrary domain; countably complete fields of sets (under set-theoretical addition); and - more generally - countably complete Boolean algebras. As examples of a less elementary character we mention cardinal numbers and relation numbers (under cardinal addition); isomorphism types of countably complete Boolean algebras and of certain more general classes of lattices (under direct multiplication); and generalized homeomorphism types of Borelian sets in an arbitrary metric space.
Several general methods are available which permit the construction of new cardinal algebras from given ones; some of them are known from modern algebra -e.g., direct multiplication and homomorphic transformation - while others are specific for the algebraic systems under discussion. A combination of these methods leads, for instance, from fields of sets to cardinal numbers, and from Boolean algebras to isomorphism types of these algebras; and hence difficult theorems on sums of cardinals and on direct products of isomorphism types appear as consequences of the fact that these and similar theorems trivially apply to set-theoretical sums of sets and to least upper bounds of elements in a Boolean algebra.
This work is divided into three parts. The first contains the definition of cardinal algebras and the development of their arithmetic. The second part is concerned with the discussion of general methods of constructing cardinal algebras; moreover, the results are extended to a wider class of algebraic systems (important from the point of view of applications), the so-called generalized cardinal algebras. In the third part we study the connections between cardinal algebras and related algebraic systems, namely, commutative semigroups and lattices; and we discuss various special examples of cardinal algebras. Certain applications of the theory of cardinal algebras to general algebra - in fact, to the theory of direct products of isomorphism types - are discussed in the appendix.
It will be seen from the above description that this work lies within the domain of abstract algebra. Some features of it, however, will probably seem strange to most algebraists - and certainly do not conform to an orthodox algebraic point of view. We have here in mind especially the part played by infinite addition and the application of frankly infinite constructions in arithmetical developments; the discussion of algebras which are not assumed to satisfy closure postulates (the generalized cardinal algebras mentioned above); and-among minor details of a rather terminological nature - the extensive use of the notion of a function.
Many arithmetical results of the first part of the work have been obtained by extending certain known theorems of the arithmetic of cardinals which originate with various authors. Some of these theorems, however, can be found only in preliminary reports, and their proof appears here for the first time. In some other cases the abstract algebraic presentation has necessitated either virtually new proofs or radical changes in old ones. The idea of an algebraic treatment of the subject and certain aspects and implications of the algebraic development seem to be essentially new.
The desire to keep the work down to a reasonable size has affected both the matter and the form of the work. Certain material which seemed to be less interesting has been entirely omitted; problems less closely related to the main idea have been merely touched upon; the arguments throughout have been presented in a rather concise form; and counter-examples have not always been supplied.
I am greatly indebted to Professors Louise H Chin (University of Arizona), Bjarni Jónsson (Brown University), and J C C McKinsey (Oklahoma Agricultural and Mechanical College) for their help in preparing this work. The assistance of Miss Chin in preparing and revising the manuscript, reading proofs, and compiling the index was indeed invaluable. As will be seen from references in footnotes, Mr Jónsson has enriched the work with a number of original contributions, and we have obtained jointly the results which are presented in the appendix.
It would be impossible for me to conclude this introduction without mentioning one more name - that of Adolf Lindenbaum, a former student and colleague of mine at the University of Warsaw. My close friend and collaborator for many years, he took a very active part in the earlier stages of the research which resulted in the present work, and the few references to his contributions that will be found in the book can hardly convey an adequate idea of the extent of my indebtedness. The wave of organized totalitarian barbarism engulfed this man of unusual intelligence and great talent - as it did millions of others. Adolf Lindenbaum was killed by the Gestapo in 1941.
University of California,
Berkeley, July 1948
JOC/EFR August 2006
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