Axiomatic Set Theory [Paperback]

The first chapter gives an informal introduction to the notion of a set, first-order predicate logic (notions of bound and free variables and quantification), and the Zermelo-Fraenkel axioms of set theory. The author describes the difficulties in the "axiom of abstraction" in the writings of Frege as pointed out by Bertrand Russell. It is pointed out that the axiom of abstraction is in fact an infinite collection of axioms, thus motivating the concept of an "axiom schema". The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite". More of the set-theoretic paradoxes are discussed, along with their classification due to F.P. Ramsey into "linguistic" and "semantical" ones.

The advantage of an older book on set theory is that more of the underlying details are explained, instead of just being formally developed. The author gives a thorough discussion of the concepts throughout the book, beginning with an organized development in chapter 2. He begins immediately with discussing the distinction between the object language and metalanguage, and the symbols to be used in the object language: constants, variables, logical connectives, quantifiers, and grouping symbols. These symbols are used to construct formulas, a subclass of which, the primitive formulas, are defined recursively, and which all formulas in the object language can be expressed in terms of. Throughout the book though the author uses additional notation that allows formulas not to be written in terms of primitive formulas. This is done to make the text more readable, but he requires that the added notion satisfy the criterion of eliminability and non-creativity. The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation. The pairing axiom, which gives the existence of a non-empty set; the sum axiom, which gives the existence of the union of a family of sets; the power set axiom, which gives the notion of the set of all subsets of a set; and the axiom of regularity, which prohibits infinite descending sequences of sets, are all discussed in detail.

Chapter 3 treats relations and functions, so important not only in mathematics but in computer science, especially in the theory of relational databases. Then in chapter 4, the author begins a study of cardinality and the cardinal numbers, proving that the finite cardinal numbers have the properties of the natural numbers, as one would expect. The author is careful to point out the need for the axiom of cardinal numbers in this study. Chapter 5 then goes into the theory of ordinal numbers, wherein it is emphasized that no special axioms are needed for the development of this theory. The author is also careful to note the special problems that arise in defining the arithmetic of natural numbers, such as defining addition recursively without using set theory. But including the apparatus of set theory does allow the replacement of the recursive definition by a proper definition. The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.

The author then shows that set theory can allow the construction of the real numbers, which takes place after the construction of the rational numbers. The famous "Dedekind cut" is discussed, along with the method of Cantor, which defines real numbers as equivalence classes of Cauchy sequences of rational numbers. The author uses the Cantor approach in the rest of the book. He also proves the famous Cantor theorem on the non-denumerability of the real numbers, and gives a brief discussion of the Continuum Hypothesis.

Chapter 8 then gives an overview of the fascinating topics of transfinite induction and ordinal number theory. Recursion theory makes its appearance again in the transfinite recursion for ordinal numbers, using the axiom schema of replacement. The non-commutativity of ordinal addition and multiplication is brought out, and the falsity of Fermat's Last Theorem and Goldbach's Hypothesis in ordinal number theory is shown. The author then shows to what extent cardinal number theory can be done without using special axioms by defining cardinal numbers as initial ordinals. The axiom of choice however is needed to show that every set has a cardinal number. The author then restates the Zermelo-Fraenkel axioms in their final form at the end of the chapter.

The final chapter gives an overview of the most controversial topic in all of set theory, if not in all of mathematics: the axiom of choice. The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy. The counterintuitive Banach-Tarski paradox is discussed, and the author shows the existence of axioms which imply the axiom of choice.

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