Introduction to Logic and to the Methodology of Deductive Sciences

• VIII •

LAWS OF ADDITION AND SUBTRACTION

47. Axioms concerning addition; general properties of operations, concepts of a group and of an Abelian group

We now turn to the second group of axioms, which consists of the following six sentences:

AXIOM 6. For any numbers y and z there exists a number x such that x = y + z; in other words: if y ϵ N and z ϵ N, then also y + z ϵ N.

AXIOM 7. x + y = y + x.

AXIOM 8. x + (y + z) = (x + y) + z.

AXIOM 9. For any numbers x and y there exists a number z such that x = y + z.

AXIOM 10. If y < z, then x + y < x + z.

AXIOM 11. If y > z, then x +y > x + z.

For the moment let us concentrate on the first four sentences of this second group, that is, Axioms 6–9. They ascribe to the operation of addition a number of simple properties which are also frequently met when considering other operations in various parts of logic and mathematics.

Special terms have been introduced to designate these properties. Thus we say that the operation O is PERFORMABLE IN THE CLASS K or that the class K is CLOSED UNDER THE OPERATION O, if the performance of the operation O on any two elements of the class K results again in an element of that same class; in other words, if, for any two elements y and z of the class K, there exists an element x of this class such that

x = y O z.

The operation O is called COMMUTATIVE IN THE CLASS K, if the result of this operation is independent of the order of the elements of the class K on which it is carried out, or, in other words, if for any two elements x and y of the class we have:

x O y = y O x.

The operation O is ASSOCIATIVE IN THE CLASS K, if the result is independent of the way in which the elements are grouped together, or, more precisely, if for any three elements x, y and z of the class the condition:

x O (y O z) = (x O y) O z

is satisfied. The operation O is said to be RIGHT-INVERTIBLE or LEFT-INVERTIBLE IN THE CLASS K, if, for any two elements x and y of the class K, there always exists an element z of the class such that

x = y O z or x = z O y,

respectively, holds. An operation O which is both right- and left-invertible is simply called INVERTIBLE IN THE CLASS K. It follows at once that a commutative operation which is right- or left-invertible must be invertible. We shall now say that a class K is a GROUP WITH RESPECT TO THE OPERATION O, if this operation is performable, associative and invertible in K; if, moreover, the operation O is commutative, the class K is called an ABELIAN GROUP WITH RESPECT TO THE OPERATION O. The concept of a group and, in particular, that of an Abelian group, forms the subject of a special mathematical discipline known as the THEORY OF GROUPS, which has already been mentioned above in Chapter V.¹

In case the class K is the universal class (or the universe of discourse of the theory considered—cf. Section 23), we usually omit the reference to this class when employing such terms as “performable”, “commutative”, and so on.

In accordance with the terminology introduced above, the Axioms 6–9 are referred to as the LAW OF PERFORMABILITY, the COMMUTATIVE LAW, the ASSOCIATIVE LAW and the LAW OF RIGHT INVERTIBILITY for the operation of addition, respectively; together they state that the set of all numbers constitutes an Abelian group with respect to addition.

48. Commutative and associative laws for a larger number of summands

Axiom 7, the commutative law, and Axiom 8, the associative law, in the form in which they have been stated here, refer to two and three numbers, respectively. But there are infinitely many other commutative and associative laws concerning more than two or three numbers. The formula:

x + (y + z) = y + (z + x),

for instance, constitutes an example of a commutative law for three summands, and the formula:

x + [y + (z + u)] = [(x + y) + z] + u

represents one of the associative laws for four summands. In addition, there are theorems of a mixed character which, generally expressed, assert that any changes in either the order or the grouping of the summands are without influence upon the result of the addition. By way of an example the following theorem may be stated.

THEOREM 9. x + (y + z) = (x + z) + y.

PROOF. By suitable substitutions we obtain from Axioms 7 and 8:

In view of (1) we may, in accordance with LEIBNIZ’S law, replace “z + y” in (2) by “y + z”; the result is the desired formula:

x + (y + z) = (x + z) + y.

In a similar manner we can derive all commutative and associative laws concerning an arbitrary number of summands from Axioms 7 and 8 together, possibly, with Axiom 6. These theorems are often used in practice in the transformation of algebraic expressions. By a transformation of an expression denoting a number we mean, as usual, an alteration of such a kind as to lead to an expression denoting the same number, which may hence be joined with the original expression by the identity sign; the expressions most frequently subjected to transformations of this kind are those which contain variables and which, therefore, are designatory functions. On the basis of the commutative and associative laws we are in a position to transform any expressions of a form such as:

x + (y + z), x + [y + (z + u)], …,

that is, expressions consisting of numerical constants and variables separated by addition signs and parentheses; in any such expression we may interchange at will both the numerical symbols and the parentheses (provided only the resulting expression has not become meaningless on account of the transposition of the parentheses).

49. Laws of monotony for addition and their converses

Axioms 10 and 11, to which we will turn now, are the so-called LAWS OF MONOTONY for addition with respect to the relations less than and greater than. We say, more generally, that the binary operation O is MONOTONIC IN THE CLASS K WITH RESPECT TO THE TWO-TERMED RELATION R, if, for any elements x, y, z of the class K, the formula:

y R z

implies:

(x O y) R (x O z),

which means that the result of performing the operation O on x and y has the relation R to the result of performing the operation O on x and z. (In the case of non-commutative operations one should, strictly speaking, differentiate between right and left monotony, the one just defined being denoted as right monotony.)

The operation of addition is monotonic not only with respect to the relations less than and greater than—a consequence of Axioms 10 and 11—but also with respect to the other relations among numbers discussed in Section 46. We shall show this here only for the relation of identity:

THEOREM 10. If y = z, then x + y = x + z.

PROOF. The sum x + y, whose existence is guaranteed by Axiom 6, is equal to itself (by Law II of Section 17):

x + y = x + y

In view of the hypothesis of the theorem, the variable “y” on the right side of this equation may be replaced by the variable “z”, and we obtain the desired formula:

x + y = x + z.

The converse of Theorem 10 is also true:

THEOREM 11. If x + y = x + z, then y = z.

We shall sketch two proofs of this theorem here. The first, based upon the law of trichotomy and Axioms 6, 10 and 11, is comparatively simple. For our later aims we require, however, another proof which is considerably more involved, but does not make use of anything except Axioms 7–9.

FIRST PROOF. Suppose the theorem in question were false. Then there would be numbers x, y and z such that

and yet

Since x + y and x + z are numbers (according to Axiom 6), they can, by the law of trichotomy, satisfy only one of the formulas:

x + y = x + z, x + y < x + z and x + y > x + z.

Since, by (1), the first holds, the others are automatically eliminated. We therefore have:

By applying the law of trichotomy once more, we can, on the other hand, infer from the inequality (2) that

y < z or y > z.

Hence, by Axioms 10 and 11,

which represents an obvious contradiction to (3). The supposition is thus refuted, and the theorem must be considered proved.

*SECOND PROOF. Apply Axiom 9, with “x” and “z” replaced by “y” and “u”, respectively. It follows that there exists a number u fulfilling the formula:

y = y + u.

Since, by Axiom 7,

y + u = u + y,

we have, on account of the transitivity of the relation of identity (cf. Law IV of Section 17):

Now apply Axiom 9 again, with “x” and “z” replaced by “z” and “v”, respectively; we thereby obtain a number v satisfying the equation:

On account of (1), we may here replace the variable “y” by the expression “u + y”:

z = (u + y) + v.

Further, by the associative law, i.e. Axiom 8, we have:

u + (y + v) = (u + y) + v,

so that, by applying Law V of Section 17, we arrive at:

z = u + (y + v).

On account of (2), we may here replace “y + v” by “z” (using LEIBNIZ’S law), so that we finally obtain:

Applying Axiom 9 for the third time, this time with “x”, “y” and “z” replaced by “u”, “x” and “w”, respectively, we obtain a number w for which

u = x + w

holds, and since

x + w = w + x,

we have:

Using (4) we obtain the following formula from (1):

y = (w + x) + y;

but since, by the associative law, we have:

w + (x + y) = (w + x) + y,

this formula becomes:

In view of the hypothesis of the theorem to be proved, we may replace “x + y” in (5) by “x + z”, which leads to:

Applying again the associative law, we have:

w + (x + z) = (w + x) + z,

so that (6) becomes:

y = (w + x) + z.

On account of (4), we may here replace “w + x” by “u”. In this way we obtain:

But from equations (7) and (3) it follows that

y = z, q.e.d.*

A few remarks concerning the first proof of Theorem 11 may be inserted here. Like the proof of Theorem 1, it constitutes an example of an indirect inference. The schema of this proof may be represented as follows. In order to prove a certain sentence, say “p”, we suppose the sentence to be false, that is we assume the sentence “not p”. From this assumption a consequence “q” is derived; that is to say, we demonstrate the implication:

if not p, then q

(in the case under consideration, the consequence “q” is the conjunction of the conditions (3) and (4) which appear in the proof). On the other hand, however, we are able to show (either on the basis of general laws of logic, as in the case under consideration, or by some theorems previously proved within the mathematical discipline in which all these arguments are carried out), that the consequence obtained is false, that is, that “not q” holds; thereby we are compelled to give up the original assumption, and thus to accept the sentence “p” as true. If this argument were set down in the form of a complete proof, a logical law which would play an essential part in it is a variant of the law of contraposition known from Section 14, and which reads as follows:

From: if not p, then q, it follows that: if not q, then p.

The proof under consideration differs slightly from that of Theorem 1. There, from the assumption that the theorem is false, we inferred that the theorem is true, that is, we derived a consequence directly contradicting the assumption; here, however, we derived from a similar assumption a consequence of which we knew from other sources that it was false. But this difference is not an essential one; it can easily be seen on the basis of logical laws that the proof of Theorem 1—like any other indirect mode of inference—can be brought under the schema sketched above.

Like Theorem 10, the other laws of monotony, that is, Axioms 10 and 11, also admit of conversion:

THEOREM 12. If x + y < x + z, then y < z.

THEOREM 13. If x + y > x + z, then y > z.

The proof of these theorems can without difficulty be obtained along the lines of the proof of Theorem 1.

50. Closed systems of sentences

There exists a general logical law the knowledge of which considerably simplifies the proofs of the last three theorems (11, 12 and 13). This law, sometimes called the LAW OF CLOSED SYSTEMS or HAUBER’S LAW², permits us in some cases, when we have succeeded in proving several conditional sentences, to infer from the form of these sentences that the corresponding converse sentences may be also considered as proved.

Suppose we are given a number of implications, say three, to which we will give the following schematic form:

if p₁, then q₁;

if p₂, then q₂;

if p₃, then q₃.

These three sentences are said to form a CLOSED SYSTEM, if their antecedents are of such a kind as to exhaust all possible cases, that is, if is true that:

p₁ or p₂ or p₃,

and if, at the same time, their consequents exclude one another:

if q₁, then not q₂; if q₁, then not q₃; if q₂, then not q₃.

The law of closed systems asserts that if certain conditional sentences forming a closed system are true, then the corresponding converse sentences are also true.

The simplest example of a closed system is given in the form of a system of two sentences, consisting of some implication:

if p, then q,

and its inverse sentence:

if not p,then not q.

In order to demonstrate the two converse sentences in this case, it is not even necessary to resort to the law of closed systems; it is sufficient to apply the laws of contraposition.

Theorem 10 and Axioms 10 and 11 form a closed system of three sentences. This is a consequence of the law of trichotomy; since between any two numbers we have exactly one of the relations =, < and >, we see that the hypotheses of these three sentences, that is, the formulas:

y = z, y < z, y > z,

exhaust all possible cases, while their conclusions, that is, the formulas:

x + y = x + z, x + y < x + z, x + y > x + z,

exclude one another. (The law of trichotomy implies even more, which however is irrelevant for our purpose, namely, that the first three formulas do not only exhaust all possible cases but also exclude each other, and that the last three formulas do not only exclude each other but also exhaust all possible cases.) For the mere reason that the three statements form a closed system it is true that the converse theorems 11–13 must hold.

Numerous examples of closed systems can be found in elementary geometry; for instance, when examining the relative position of two circles, we have to deal with a closed system consisting of five sentences.

In conclusion it may be remarked that anyone who does not know the law of closed systems but tries, to prove the converses of statements forming a system of this kind may mechanically apply the same mode of inference which we employed in the first proof of Theorem 11.

51. Consequences of the laws of monotony

Theorems 10 and 11 may be combined into one sentence:

y = z if, and only if, x + y = x + z.

We will derive one more consequence from the theorems of monotony:

THEOREM 14. *If x + z < y + t, then x < y or z < t.

PROOF. Suppose the conclusion of the theorem is false; in other words, neither is x smaller than y nor is z smaller than t. From this it follows by the law of trichotomy that one of the two formulas:

x = y or x > y

and also one of the two formulas:

z = t or z > t

must hold. We thus have to discuss the following four possibilities:

Let us begin by considering the first case. If the two equations (1) are valid, we obtain, by Theorem 10:

z + x = z + y

from the first equation; and since, according to Axiom 7,

x + z = z + x and z + y = y + z

we may infer, by a twofold application of the law of transitivity for the relation of identity:

If now we apply Theorem 10 to the second of the equations (1), we obtain:

which, together with (5), yields:

By an entirely analogous inference—applying Axioms 4, 5, 10 and 11—any of the three remaining cases (2), (3) and (4) lead to the inequality:

One of the formulas (7) or (8) must therefore hold in any case. But since x + z and y + t are numbers (Axiom 6), it follows by the law of trichotomy that the formula:

x + z < y + t

cannot hold.

Thus, by assuming the conclusion to be false, we have arrived at an immediate contradiction to the hypothesis of the theorem. The assumption is therefore to be refuted, and we see that the conclusion does indeed follow from the hypothesis.

The argument just conducted is counted among the indirect proofs; apart from an inessential modification, it could be brought under the schema sketched in Section 49 in connection with the first proof of Theorem 11. Formally considered, however, the procedure of the argument is slightly different from the one followed in the proofs of Theorems 1 and 11. The inference has the following schema. In order to prove a sentence of the form of an implication, say, the sentence:

if p, then q,

we assume the conclusion of the sentence, that is, “q”, to be false (and not the whole sentence); from this assumption, that is, from “not q”, it is inferred that the hypothesis is false, that is, that “not p” holds. In other words, instead of demonstrating the sentence in question, a proof of the corresponding contrapositive sentence:

if not q, then not p

is given, and from this the validity of the original sentence is inferred. The basis for an inference of this kind is to be found in a law of sentential calculus to the effect that the truth of the contra-positive sentence always implies that of the original sentence (cf. Section 14).

Inferences of this form are very common in all mathematical disciplines; they constitute the most usual type of indirect proof.

52. Definition of subtraction; inverse operations

Our next task is to show how the notion of subtraction can be introduced into our considerations. With this aim in mind, we shall first prove the following theorem:

THEOREM 15. For any two numbers y and z there is exactly one number x such that y = z + x.

PROOF. Axiom 9 guarantees the existence of at least one number x satisfying the formula:

y = z + x.

We have to show that there is no more than one such number; in other words, that any two numbers u and v satisfying this formula are identical. Let, therefore,

y = z + u and y = z + v.

This implies at once (by the laws of symmetry and transitivity for the relation =):

z + u = z + v,

from which, by Theorem 11, we obtain:

u = v.

There is, thus, exactly one number x (cf. Section 20) for which

y = z + x, q.e.d.

This unique number x, of which the above theorem treats, is designated by the symbol:

y − z;

we read it, as usual, “the difference of the numbers x and y” or “the result of subtracting the number z from the number y”. The precise definition of the notion of difference is as follows:

DEFINITION 2. We say that x = y − z if, and only if, y = z + x.

An operation I is called a RIGHT INVERSE OF THE OPERATION O IN THE CLASS K if these two operations O and I fulfil the following condition:

for any elements x, y and z of the class K, we have: x = y I z if, and only if, y = z O x.

The analogous concept of a LEFT INVERSE OF THE OPERATION O is defined similarly. If the operation O is commutative in the class K, its two inverses—the right and the left—coincide, and we can then simply speak of the INVERSE OF THE OPERATION O (or, also, of the INVERSE OPERATION OF O). In accordance with this terminology, Definition 2 expresses the fact that subtraction is the right inverse (or, simply, the inverse) of addition.

53. Definitions whose definiendum contains the identity sign

*Definition 2 exemplifies a kind of definition very common in mathematics. These definitions stipulate the meaning of a symbol designating either a single thing or an operation on a certain number of things (in other words, a function with a certain number of arguments). In every definition of this kind, the definiendum has the form of an equation:

x = …;

on the right side of this equation, we have the symbol itself which was to be defined, or else a designatory function constructed out of the symbol to be defined and certain variables “y”, “z”, …, according as the symbol in question designates a single thing or an operation on things. The definiens may be a sentential function of any form, which contains the same free variables as the definiendum, and which states that the thing x—together possibly with the things y, z, …—satisfies such and such a condition.—Definition 2 establishes the meaning of a symbol which denotes an operation on two numbers. To give a different example of this type of definition, let us state the definition of the symbol “0” which designates a single number:

we say that x = 0 if, and only if, for any number y, the formula: y + x = y holds.

A certain danger is connected with definitions of the type under consideration; for if one does not proceed with sufficient caution in laying down such definitions, one can easily find oneself confronted with a contradiction. A concrete example will make this clear.

Let us leave, for the moment, our present investigations, and assume that in arithmetic we have already the symbol of multiplication at our disposal and that, with its help, we want to define the symbol of division. For this purpose we proceed to lay down the following definition, which is modelled precisely after Definition 2:

we say that x = y : z if, and only if, y = z · x.

If now, in this definition, we replace both “y” and “z” by “0”, and “x” first by “1” and then by “2”, and if we observe that we have the formulas:

0 = 0.1 and 0 = 0.2,

we obtain at once:

1 = 0:0 and 2 = 0:0.

But since two things equal to the same thing are equal to each other, we arrive at:

1 = 2,

which is obviously nonsense.

It is not hard to exhibit the reason for this phenomenon. Both in Definition 2 and in the definition of the quotient considered here, the definiens has the form of a sentential function with three free variables “x”, “y” and “z”. To each such sentential function there corresponds a three-termed relation holding between the numbers x, y and z if, and only if, these numbers satisfy that sentential function (cf. Section 27); and it is just the aim of the definition to introduce a symbol designating this relation. But if one gives the definiendum the form:

x = y − z or x = y : z,

one assumes in advance that this relation is functional (and hence an operation, or a function, cf. Section 34), and that therefore, to any two numbers y and z, there is at most one number x standing to them in the relation in question. The fact that the relation is functional, however, is not at all evident from the beginning, and it must first be established. This we did in the case of Definition 2; but we failed to do so in the case of the definition of the quotient, and we would indeed have been unable to do so, simply because the relation in question ceases to be functional in a certain exceptional case: for, if

y = 0 and z = 0,

there exist infinitely many numbers x for which

y = z · x.

If, therefore, one wants to formulate the definition of the quotient in the above form without introducing contradictions, one has to take care that the case is excluded where both numbers y and z are 0,—for instance, by inserting an additional condition in the definiens.

The above considerations lead us to the following conclusion. Every definition of the type of Definition 2 should be preceded by a theorem corresponding exactly to Theorem 15, that is to say, a theorem to the effect that there is but one number x which satisfies the definiens. (The question arises whether it is relevant if there is exactly one number x, or whether it is sufficient that there is at most one such number. A discussion of this rather difficult problem will be omitted here.)*

54. Theorems on subtraction

On the basis of Definition 2 and the laws of addition we can without difficulty prove the fundamental theorems of the theory of subtraction, such as the law of performability, the laws of monotony, and the laws of equivalent transformation of equations and inequalities by means of subtraction. Those theorems also belong here which make possible the transformation of so-called algebraic sums, that is, of expressions consisting of numerical constants and variables, separated by “+” and “−” signs as well as parentheses (the latter often being omitted in accordance with special rules to this effect). The following theorem may serve as an example of the last-named category:

THEOREM 16. x + (y − z) = (x + y) − z.

PROOF. To y and z, according to Axiom 9, there corresponds a number u such that

this implies, by Definition 2,

From the commutative law we have:

x + y = y + x.

On account of (1), “y” may here be replaced by “z + u” on the right side, so that we obtain:

From Theorem 9, on the other hand, it follows that:

But since two numbers equal to the same number are equal to each other, we can infer from (3) and (4):

Now, since x + u and x + y are numbers (by Axiom 6), we may substitute “x + u” and “x + y” for “x” and “y” in Definition 2. (5) shows that the definiens is then satisfied, and hence the definiendum must also hold:

x + u = (x + y) − z.

If now, in view of (2), we replace “u” by “y − z” in this last equation, we finally arrive at:

x + (y − z) = (x + y) − z, q.e.d.

Having gotten this far, we now terminate the construction of our fragment of arithmetic.

Exercises

1. Consider the following three systems, each consisting of a certain set, two relations and one operation:

(a) the set of all numbers, the relations and , the operation of addition;

(b) the set of all numbers, the relations < and >, the operation of multiplication;

Determine which of these systems are models of the system of Axioms 1–11 (cf. Section 37).

2. Consider an arbitrary straight line, to which we will refer as the number line; let the points on this line be denoted by the letters “X”, “Y”, “Z”, …. On the number line we choose a fixed initial point O and a unit point U distinct from O. Now let X and Y be any two distinct points on our line. We consider the two half-lines, one beginning at O and going through U, the other beginning at X and going through Y. We shall say that the point X precedes the point Y, in symbols:

if, and only if, either the two half-lines are identical or one of them—no matter which—is a part of the other. In the same situation we shall also say that the point Y succeeds the point X, written:

The point Z is called the sum of the points X and Y if it fulfils the following conditions: (i) the segment OX is congruent to the segment YZ; (ii) if , then , but if , then . The sum of the points X and Y is denoted by:

X ⊕ Y.

Show by means of the theorems of geometry that the set of all points of the number line (that is, more simply, the number line itself), the relations and , and the operation ⊕, together form a model of the axiom system adopted by us, and that, therefore, this system has an interpretation within geometry.

3. Let us consider four operations A, B, G and L which—like addition—correlate a third number with any two numbers. As the result of the operation A on the numbers x and y we always consider the number x, and as the result of the operation B the number y:

x A y = x, x B y = y.

By the symbols “x G y” and “x L y” we denote that of the two numbers x and y which is not less than or not greater than the other, respectively; we thus have:

x G y = x and x L y = y in case that x y;
x G y = y and x L y = x in case that x y.

Which of the properties discussed in Section 47 belong to these four operations? Is the set of all numbers a group and, in particular, an Abelian group with respect to any of these operations?

4. Let C be the class of all point sets, that is, of all geometrical configurations. Are the addition and multiplication of sets (as defined in Section 25) performable, commutative, associative and invertible in the class C? Is, therefore, the class C a group and, in particular, an Abelian group with respect to any of these operations?

5. Show that the set of all numbers is not an Abelian group with respect to multiplication, but that every one of the following sets is an Abelian group with respect to that operation:

(a) the set of all numbers different from 0;

(b) the set of all positive numbers;

6. Consider the set S consisting of the two numbers 0 and 1, and let the operation ⊕ on the elements of this set be defined by the following formulas:

0 ⊕ 0 = 1 ⊕ 1 = 0,

0 ⊕ 1 = 1 ⊕ 0 = 1.

Determine whether the set S is an Abelian group with respect to the operation ⊕.

7. Consider the set S consisting of the three numbers 0, 1 and 2. Define an operation ⊕ on the elements of this set, so that the set S will be an Abelian group with respect to this operation.

8. Prove that no set consisting of two or three different numbers can be an Abelian group with respect to addition. Is there a set consisting of one single number that forms an Abelian group with respect to addition?

9. Derive the following theorems from Axioms 6–8:

(a) x + (y + z) = (z + x) + y;

(b) x + [y + (z + t)] = (t + y) + (x + z).

10. How many expressions can be obtained from each of these expressions:

x + (y + z), x + [y + (z + t)], x + {y + [z + (t + u)]}

if they are transformed solely on the basis of Axioms 6–8?

11. Formulate the general definition of left monotony of an operation O with respect to a relation R.

12. On the basis of the axioms adopted by us and the theorems derived from them, prove that addition is a monotonic operation with respect to the relations ≠, and .

13. Is multiplication a monotonic operation with respect to the relations < and >

(a) in the set of all numbers,

(b) in the set of all positive numbers,

14. Which of the operations defined in Exercise 3 are monotonic with respect to the relations =, <, >, ≠, and ?

15. Are the addition and multiplication of classes monotonic with respect to the relation of inclusion? Or with respect to any of the other relations among classes discussed in Section 24?

16. Derive from our axioms the following theorem:

if x < y and z < t, then x + z < y + t.

Replace in this sentence the symbol “<” in turn by “>”, “=”, “≠”, “” and “”, and examine whether the sentences obtained in this way are true.

17. Give examples of closed systems of sentences within arithmetic and geometry.

18. Derive the following theorems from our axioms:

(a) if x + x = y + y, then x = y;

(b) if x + x < y + y, then x < y;

Hint: Prove the converse sentences first (using the results of Exercise 16), and show that they form a closed system.

*19. If a theorem is derivable from Axioms 6–9 alone, it can be extended to arbitrary Abelian groups, since every class K which forms an Abelian group with respect to an operation O constitutes, together with this operation, a model of Axioms 6–9 (cf. Sections 37 and 38). This applies, in particular, to Theorem 11 (in view of the second proof of this theorem), and we have the following general group-theoretical theorem:

every class K which is an Abelian group with respect to the operation O satisfies the following condition:

if , , and x O y = x O z, then y = z.

Give a strict proof of this theorem.

Show, on the other hand, that Theorem (a) of Exercise 18 cannot be extended to arbitrary Abelian groups, by exhibiting an example of a class K and an operation O with these properties: (i) the class K is an Abelian group with respect to the operation O, and (ii) there exist two distinct elements x and y of the class K for which x O x = y O y (cf. Exercise 6). Consequently, is it possible to derive Theorem (a) from Axioms 6–9 alone?

20. Transform the proof of Theorem 14 in such a manner that it conforms to the schema sketched in Section 49 in connection with the first proof of Theorem 11.

21. May the operation of division be said to be the inverse of multiplication in the set of all numbers?

22. Do the operations mentioned in Exercises 3 and 4 possess inverses (in the set of all numbers, or in the class of all geometrical configurations)?

23. What operations are the left and the right inverses of subtraction (in the set of all numbers)?

*24. In Section 53, the definition of the symbol “0” was stated by way of an example. In order to be certain that this definition does not lead to a contradiction, it should be preceded by the following theorem:

there exists exactly one number x such that, for any number y, we have: y + x = y.

Prove this theorem on the basis of Axioms 6–9 alone.

25. Formulate the sentences which assert that subtraction is performable, commutative, associative, right- and left-invertible, and right- and left-monotonic with respect to the relation less than. Which of these sentences are true? Prove those for which this is the case, using our axioms and Definition 2 of Section 52.

26. Derive the following theorems from our axioms and Definition 2:

(a) x − (y + z) = (x − y) − z,

(b) x − (y − z) = (x − y) + z,

*27. Using the law of performability for subtraction and Theorem (c) of the preceding exercise, prove the following theorem:

for a set K of numbers to be an Abelian group with respect to addition, it is necessary and sufficient that the difference of any two numbers of the set K also belongs to the set K (i.e. that the formulas and always imply ).

Use this theorem in order to find examples of sets of numbers that are Abelian groups with respect to addition.

28. Write in logical symbolism all axioms, definitions and theorems given in the last two chapters.

Hint: Before formulating Theorem 15 in symbols, put it in an equivalent form in which the numerical quantifiers have been eliminated by virtue of the explanations given in Section 20.

¹ The group concept was introduced into mathematics by the French mathematician E. GALOIS (1811–1832). The term “Abelian group” was chosen in honor of the Norwegian mathematician N. H. ABEL (1802–1829) whose researches have had a great influence upon the development of higher algebra. The far-reaching importance of the group concept for mathematics has been recognised particularly since the works of another Norwegian mathematician, S. LIE (1842–1899).

² After the name of the German mathematician K. F. HAUBER (1775–1861).