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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.
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