Logic for mathematicians.

*(English)*Zbl 0068.00707
International Series in Pure and Applied Mathematics. New York-Toronto-London: McGraw-Hill Book Company, Inc. xiv, 530 p. (1953).

This book falls into two parts of which the first, comprising chapters I–IX, contains a rather broad treatment of logic, adapted in an original way to the needs of mathematicians by the insertion of numerous applications. Almost every logical theorem of any importance is thus illustrated by an example, taken from some mathematical textbook. Of course the complexity of these applications increases with the construction of the logical building; they range from simple examples taken from Euclid to a treatment of open and closed sets in Hausdorff space as an application of the logic of classes.

Ch. I is introductory. In Ch. II the statement calculus is treated from an intuitive point of view and the method of truth tables is introduced. Ch. III, “The use of names”, stresses the difference between a thing and its name. In Ch. IV the statement calculus is treated by the axiomatic method, and the completeness theorem is proved.

In Ch. V. “Clarification”, it is stated as the aim of the book to set up a symbolic logic such that none of its principles is known to be invalid (no definition of this term is attempted) and from which the existing body of mathematics can be derived. Some remarks are made about the modes of reasoning which are allowed in metalogic, but no adequate description of the finitistic point of view is given.

Ch. VI contains the predicate calculus of first order. The axioms are given in closed form, so that the only derivation rule needed is modus ponens. Attention is given to restricted quantification: if \(\alpha\) is a variable restricted to the objects satisfying \(K(x)\), then \((\alpha) F(\alpha)\) is an abbreviation for \((x) \cdot K(x) \supset F(x)\). In Ch. VII equality is introduced, and in Ch. VIII the notation \(ix F(x)\) for (the \(x\) such that \(F(x))\).

Ch. IX contains the logic of classes. It begins with a discussion of the notion of class and of the Russell paradox. Various proposals for eliminating the paradox are examined (here the description of Brouwer’s point of view on p. 203–204 is erroneous). The system adopted for the book is that of Quine’s “New foundations” (this Zbl. 16, 193); consequently, many theorems are subject to conditions of stratification. The formal system is now more precisely described; here equality is defined in terms of class membership and a system of 12 axiom schemes is given (p. 212–213). Additional axioms are introduced in the following chapters; they will be mentioned below. A section is devoted to variables restricted to a fixed class \(\Sigma\); here a division analogous to the hierarchy of types is obtained.

In the second part of the book the program of Frege, Russell and Whitehead, the building up of mathematics on the basis of logic, is taken up with all the simplifications which later research has made possible. Here also the author keeps in contact with mathematical literature, from which he borrows examples to the main theorems.

Ch. X, “Relations and functions”, begins with the theory of natural numbers \(Nn\). In sharp contrast with the ample explanations of other fundamental concepts, Frege’s definition is given without any comment. The principle of mathematical induction is proved for stratified conditions, and Peano’s fourth axiom is introduced as Axiom scheme 13, equivalent to the axiom of infinity The calculus of relations is developed. There is an interesting discussion of the ambiguity of the word “function” in mathematics. The author uses “function” in the sense of “many-one-relation”. Order relations are treated: a section on equivalence relations leads over to Ch. XI on cardinal numbers \(NC\). Of course it cannot be proved that any class is similar to the class of its unit subclasses; a class with this property is called cantorian. The arithmetic of cardinal numbers is treated and it is proved that \(Nn \subset NC\); a class is called finite if its cardinal number is a \(Nn\). Various forms of the induction principle are discussed. As to definition by induction it is argued that intuitively it defines the function value for any natural number, but not the function itself. A formal proof is given for the latter, under a stratification condition. No finite class can be similar to a proper subclass of itself, but the converse is proved only by making use of the axiom of choice (Ch. XIV). In the next section it is proved among other things that \(Nn\) is cantorian. Finally, the arithmetical properties of the cardinal number of the continuum are proved.

Ch. XII is on ordinal numbers. In Ch. XIII, “Counting”, the author observes that for any fixed natural number \(n\) the formula \(A(n), m\) \((m\in Nn\cdot 0 < m < n) \in n\), can be proved, but no proof of \((n):n\in Nn\cdot\supset \cdot A(n)\) is known. Thus he proposes the latter as Axiom scheme 14. It allows one to prove that every finite class is cantorian.

Ch. XIV is on the axiom of choice. The equivalence of this axiom with Zermelo’s well-ordering theorem and with various forms of Zorn’s lemma is proved. The denumerable axiom of choice is proposed as Axiom scheme 15; it is used to prove that every infinite class is similar to a proper subclass of itself, and that the union of a denumerable class of denumerable classes is denumerable.

In the concluding Ch. XV a reading program is given with the intention of showing that the notions introduced in the book suffice to develop the bulk of modern mathematics. As it was impossible to summarize the formal deductions which form the main content of the book, this review stresses disproportionately some controversial points.

The author has succeeded in giving a modern treatment of the subject matter of “Principia mathematica” (2nd ed., 3 vols., Cambridge, 1925, 1927); at the same time he has written a book of great didactic value.

Ch. I is introductory. In Ch. II the statement calculus is treated from an intuitive point of view and the method of truth tables is introduced. Ch. III, “The use of names”, stresses the difference between a thing and its name. In Ch. IV the statement calculus is treated by the axiomatic method, and the completeness theorem is proved.

In Ch. V. “Clarification”, it is stated as the aim of the book to set up a symbolic logic such that none of its principles is known to be invalid (no definition of this term is attempted) and from which the existing body of mathematics can be derived. Some remarks are made about the modes of reasoning which are allowed in metalogic, but no adequate description of the finitistic point of view is given.

Ch. VI contains the predicate calculus of first order. The axioms are given in closed form, so that the only derivation rule needed is modus ponens. Attention is given to restricted quantification: if \(\alpha\) is a variable restricted to the objects satisfying \(K(x)\), then \((\alpha) F(\alpha)\) is an abbreviation for \((x) \cdot K(x) \supset F(x)\). In Ch. VII equality is introduced, and in Ch. VIII the notation \(ix F(x)\) for (the \(x\) such that \(F(x))\).

Ch. IX contains the logic of classes. It begins with a discussion of the notion of class and of the Russell paradox. Various proposals for eliminating the paradox are examined (here the description of Brouwer’s point of view on p. 203–204 is erroneous). The system adopted for the book is that of Quine’s “New foundations” (this Zbl. 16, 193); consequently, many theorems are subject to conditions of stratification. The formal system is now more precisely described; here equality is defined in terms of class membership and a system of 12 axiom schemes is given (p. 212–213). Additional axioms are introduced in the following chapters; they will be mentioned below. A section is devoted to variables restricted to a fixed class \(\Sigma\); here a division analogous to the hierarchy of types is obtained.

In the second part of the book the program of Frege, Russell and Whitehead, the building up of mathematics on the basis of logic, is taken up with all the simplifications which later research has made possible. Here also the author keeps in contact with mathematical literature, from which he borrows examples to the main theorems.

Ch. X, “Relations and functions”, begins with the theory of natural numbers \(Nn\). In sharp contrast with the ample explanations of other fundamental concepts, Frege’s definition is given without any comment. The principle of mathematical induction is proved for stratified conditions, and Peano’s fourth axiom is introduced as Axiom scheme 13, equivalent to the axiom of infinity The calculus of relations is developed. There is an interesting discussion of the ambiguity of the word “function” in mathematics. The author uses “function” in the sense of “many-one-relation”. Order relations are treated: a section on equivalence relations leads over to Ch. XI on cardinal numbers \(NC\). Of course it cannot be proved that any class is similar to the class of its unit subclasses; a class with this property is called cantorian. The arithmetic of cardinal numbers is treated and it is proved that \(Nn \subset NC\); a class is called finite if its cardinal number is a \(Nn\). Various forms of the induction principle are discussed. As to definition by induction it is argued that intuitively it defines the function value for any natural number, but not the function itself. A formal proof is given for the latter, under a stratification condition. No finite class can be similar to a proper subclass of itself, but the converse is proved only by making use of the axiom of choice (Ch. XIV). In the next section it is proved among other things that \(Nn\) is cantorian. Finally, the arithmetical properties of the cardinal number of the continuum are proved.

Ch. XII is on ordinal numbers. In Ch. XIII, “Counting”, the author observes that for any fixed natural number \(n\) the formula \(A(n), m\) \((m\in Nn\cdot 0 < m < n) \in n\), can be proved, but no proof of \((n):n\in Nn\cdot\supset \cdot A(n)\) is known. Thus he proposes the latter as Axiom scheme 14. It allows one to prove that every finite class is cantorian.

Ch. XIV is on the axiom of choice. The equivalence of this axiom with Zermelo’s well-ordering theorem and with various forms of Zorn’s lemma is proved. The denumerable axiom of choice is proposed as Axiom scheme 15; it is used to prove that every infinite class is similar to a proper subclass of itself, and that the union of a denumerable class of denumerable classes is denumerable.

In the concluding Ch. XV a reading program is given with the intention of showing that the notions introduced in the book suffice to develop the bulk of modern mathematics. As it was impossible to summarize the formal deductions which form the main content of the book, this review stresses disproportionately some controversial points.

The author has succeeded in giving a modern treatment of the subject matter of “Principia mathematica” (2nd ed., 3 vols., Cambridge, 1925, 1927); at the same time he has written a book of great didactic value.

Reviewer: A. Heyting (M. R. 14 # 935)

##### MSC:

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |