Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
A SET OF POSTULATES FOR ARITHMETIC AND ALGEBRA B Y HENRY BLUMBERG. In the following paper, I wish to present a set of postulates for arithmetic and algebra that, I believe, combines very well the advantages of the so-called " genetic " and " axiomatic " methods for foundations in mathematics. This set has grown out of conversations that I had with Prof. Zermelo and it is our intention to publish our results in greater detail in the near future. To-day I shall confine myself to indicating the principal points. We start with a set of postulates defining the most general field ("Körper") for which the commutative law of multiplication need not hold. Such a field will be henceforth simply termed a "non-commutative" field. By the addition of one postulate, we obtain a second set of postulates defining the most general non-commutative field that contains as a sub-field the field of rational numbers'*. By the further addition of one postulate, we obtain a third set defining the most general non-commutative field that contains as a sub-field the field of real numbers. Finally, by the addition of one postulate to this third set, we obtain a fourth set of postulates defining the most general non-commutative field that contains as a sub-field the field of ordinary complex numbers. Certain concepts and principles of logic and the theory of aggregates will be presupposed without being expressly formulated. The notion of " finite number " is however not presupposed. In the proofs, therefore, it will not be permissible to use the expression " a finite number of times " or the expression " and so on " when reference is implicitly made to the series of ordinal numbers. The development of arithmetic and algebra on the basis of our postulates includes, therefore, a rigorous development of the theory of finite numbers. In this connection, I may refer to a paper of Prof. Zermelo, read at the Fourth International Congress of Mathematicians and entitled, " Ueber die Grundlagen der Arithmetik." I also wish to call attention to the fact that there are no postulates of order. For other sets of postulates for arithmetic and algebra to be found in mathematical literature, I refer to Hilbert's Grundlagen der Geometrie (third edition) and, in particular, to the articles of Prof. Huntington in the Transactions of the American Mathematical Society and in the Annals of Mathematics. In these articles of Huntington you will also find extensive bibliographies. * More accurately, "that contains as a sub-field a field isomorphic with the field of rational numbers with respect to addition and multiplication." A similar remark is to be made later with regard to the real and the complex numbers. 462 HENRY BLUMBERG I may add, by way of explanation, that the term "equality" has the same significance below as the term " identity." The logical properties of the latter concept are presupposed. Let S be an aggregate of objects; "addition," denoted by the symbol +, and "multiplication," denoted by the symbol x, two operations. The scheme of our postulates is then as follows : Postulate Ax. element of g. If a and b are elements of %, then a + b is a uniquely determined Postulate A2. The associative law of addition holds for § ; i.e. if a, b, c are elements of %, then (a + b) + c = a + (b + c). Postulate As. There exists an element z (zero) in %, such that for every element a of g (1) a + z = a, and (2) an element ä exists such that a + ä = z. Postulate Mx. If a and b are elements of %, then a x b is a uniquely determined element of g. Postulate M2. The associative law of multiplication holds for % ; i.e. if a, b, c are elements of 5, then (a x b) x c = a x (b x c). Postulate Ms. There exists an element u (unity) in g, such that for every element a of 5 that satisfies the inequality a + a =j= a (1) axu = a, and (2) an element a' exists such that ax a =u. Postulate D. The distributive laws of addition and multiplication hold for g ; i.e. if a, b, c are elements of %, then a x (b + c) = (a x b) + (a x c) and (b + c) x a = (b x a) + (c x a). These seven postulates constitute the first set of postulates mentioned above ; i.e. they define the most general non-commutative field. Among the important deductions from these postulates are : (1) The uniqueness of the zero element z and the unit element u. (2) The uniqueness of the inverse elements ä and a of an element a, the former with respect to addition and the latter with respect to multiplication. (3) The commutative law of addition. (4) The law that a product equals z when and only when at least one of its factors equals z. I now wish to indicate how the integral and rational numbers are defined and ordered. Definition : An ^-aggregate is a system of elements of g that contains (a) the element u, (ß) with every element x the element x + u. That such aggregates exist is shown by the fact that § itself is an i^-aggregate. A SET OF POSTULATES FOR ARITHMETIC AND ALGEBRA 463 The greatest common divisor of all the ^/-aggregates is again an ^-aggregate. This special i?-aggregate we denote by X. Its elements play the rôle of natural or positive integral numbers. X may also be defined as follows : X is an /^-aggregate such that every sub-aggregate of it that is also an ^-aggregate is identical with X. For X the principle of mathematical induction holds. formulated as follows : Principle of induction for X: object x such that and This principle may be If Ax is a statement regarding an undetermined (a) Au is true, (ß) the truth of Ax implies the truth of Ax+U, then Ax is true for every element x of N. By means of this principle, w^e can prove the Theorem : If a and b belong to N, then a + b and axb belong to N. The following theorem holds : Theorem : N is infinite or not according as z is not or is contained in N. positive definition of infinity by means of correspondence is employed.) (The Definition : SR is the greatest common divisor of all non-commutative fields that are sub-aggregates of g. (8 itself is also to be considered a sub-aggregate of $.) The following theorem holds : Theorem : If X is not infinite, it is identical with 3Ì. We are now in a position to appreciate the meaning of the next postulate. Posttdate R. Every non-commutative sub-field of % is infinite. By virtue of this postulate, we obtain on the basis of the preceding theorems the Theorem : z is not contained in X. From this follows the Theorem : If a is contained in X, then ci is not contained in X. We now define the aggregate G as follows : An element x of % belongs then and only then to G, when either x = z or x belongs to N or x belongs to X. The following theorem holds : Theorem : The sum, difference and product of any two elements of G also belong to G. We are now ready to introduce the notion of order for the elements of G. When the two elements a and b of G are such that a + b belongs to N, we say, " a is greater than b " and write, " a > b." It may then be shown that for any two elements a and b of G, one and only one of the three relations, a>b, a = b, b>a, holds. All the wellknown theorems of order for integral numbers may then be deduced by the aid of the foregoing theorems. In particular, for example, the fundamental theorem may be proved, that every sub-aggregate of X possesses a smallest element. 464 HENRY BLUMBERG We then show : Theorem : Every element of?fc has the form m x n, where m and n belong to N. By means of this theorem we order the elements of 9i on the basis of the order of the elements of X. By means of the last theorem we are also able to prove the Theorem : If a is any element of % and r any element of dì, then ax r = r x a. In particular, then, the commutative law of multiplication holds for dt. We thus see that these eight postulates define the most general non-commutative field containing as a sub-field the field of rational numbers. If it is our desire to obtain a set of postulates defining the rational numbers alone, we need only add the following postulate to those already given. Postulate P. g is primitive with respect to the foregoing postulates ; i.e. every sub-aggregate of % for which the foregoing postulates hold is identical with %. (This postulate may be called the " postulate of primitivity.") It may be remarked, in passing, that on the supposition of the consistency of the postulates AY—As, M1—Ms, D and R, the consistency of the entire set of nine postulates may be proved. For, by means of the first eight, we were able to show the existence of dt, which in fact satisfies all of our nine postulates. In the ordered aggregate di, we may define a Dedekind's section in the usual way. The sum and the product of two sections are then also defined, as certain definite sections, in the usual way. Our next postulate is as follows : Postulate C. To every section S of dt there corresponds uniquely an element s of g such that, when Sj and S2 are two sections and s1 and s2 the corresponding elements, s1 + s2, s1x s2 correspond respectively to S1 + S2, S1xS2; the corresponding elements are furthermore not all identical. The set of postulates AY—A3, M1—M3, D, R and C defines the most general non-commutative field containing as a sub-field the field of real numbers. If we denote by (J the aggregate that consists of dt and all the elements of g that correspond to sections of dt, it may be shown, after the manner of Dedekind, that S possesses all the properties of the ordinary real numbers. In particular, I mention the fact that the commutative law of multiplication can be proved to hold for S. To obtain a set of postulates defining the real numbers alone, we need only add to the 9 postulates Ax—A3, MY—Ms, D, R and C the postulate of primitivity P unaltered. To obtain a set of postulates for the most general non-commutative field containing as a sub-field the field of ordinary complex numbers, we add to the postulates already given (the postulate of primitivity being of course excepted) the following : Postulate I. There exists in g an element i such that ixi element c of $ i x c = c x i. = u and for every I t may be shown that the requirement that i x c = c x i for every element c of ß is not redundant. A SET OF POSTULATES FOR ARITHMETIC AND ALGEBRA 465 In a similar manner as above, we obtain a set of postulates for complex numbers alone by adding to the foregoing postulates the postulate of primitivity P. The field of quaternions may also be dealt with from our point of view ; but I do not wish to enter upon its discussion here. I also do not wish to discuss the question of independence. however essentially independent. Our postulates are To sum up, I append a list of the sets of postulates considered. For the most general non-commutative field : Al9 A2, A3, Mlt M2, M3, D. For the most general non-commutative field containing rational numbers: A,, A2, A3, Ml9 M2, M3, D, R. For the most general non-commutative field containing real numbers: Aly A2, ii,, Ml9 M2, Ms, D, R, C. For the most general non-commutative field containing complex numbers: Aly A2, A3, M1} M2, M3, D, R, C, as a sub-field the field of as a sub-field the field of as a sub-field the field of I. For the field of rational numbers : Au A2, As, Mu M2, M3i D, R, P. For the field of real numbers : A1} A2, A3, Mly M2, Ms, D, R, C, P. For the field of complex numbers : A,, A2, As, Mu M2) M3i D R, C, I, P. M. C. II. 30