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NUMA POMPILIUS
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NUMA POMPILIUS , second legendary See also: The supposed law-books, which were to all See also:appearance new when discovered, were clearly forgeries. See Livy i. 18-21; See also:Plutarch, Numa; See also:Dion. Halic. ii. 58-76; See also:Cicero, De republica, ii. 13-15. For See also:criticism: See also:Schwegler, Romische Geschichte, bk. xi.; See also:Sir G. Cornewall See also:Lewis, Credibility of early Roman See also:History, ch. xi.; W. Ihne, Hist. of Rome, i.; E. Pais, Storia di See also:Roma, i. (1898), where Numa is identified with Titus Tatius and made out to be a See also:river god, Numicius, closely connected with See also:Aeneas; J. B. See also:Carter, The Religion of Numa (1906); O. See also: 2. Aggregates (also called manifolds or sets).— Let us assume the possibility of constructing or contemplating a permanent system of things such that (1) the system includes all See also:objects to which a certain definite quality belongs; (2) no See also:object without this quality belongs to the system; (3) each object of the system is permanently recognizable as the same thing, and as distinct from all other objects of the system. Such a collection is called an aggregate: the See also:separate objects belonging to it are called its elements. An aggregate may consist of a single See also:element. It is further assumed that we can select, by a definite process, one or more elements of any aggregate A at See also:pleasure: these See also:form another aggregate B. If any element of A remains unselected, B is said to be a See also:part of A (in symbols, B < A) : if not, B is identical with A. Every element of A is a part of A. If B< A and C-< B, then C< A. When a See also:correspondence can be established between two aggregates A and B in such a way that to every element of A corresponds one and only one element of B, and conversely, A and B are said to be See also:equivalent, or to have the same See also:power (or potency) ; in symbols, A B. If A w B and B co C, then A co C. It is possible for an aggregate to be equivalent to a part of itself: the aggregate is then said to be See also:infinite. As an example, the aggregates 2, 4, 6, . . . an, &c., and 1, 2, 3, . . . n, &c., are equivalent, but the first is only a part of the second. 3. See also:Order.—Suppose that when any two elements a, b of an aggregate A are taken there can be established, by a definite criterion, one or other of two alternative relations, symbolized by a<b and a>b, subject to the following conditions: (r) If a>b, then b<a, and if a<b, then b>a; (2) If a>b and b>c, then a>c. In this See also:case the criterion is said to arrange the aggregate in order. An aggregate which can be arranged in order may be called ordinable. An ordinable aggregate may, in See also:general, by the application of different criteria, be arranged in order in a variety of ways. According as a<b or a>b we shall speak of a as anterior or posterior to b. These terms are chosen merely for convenience, and must not be taken to imply any meaning except what is involved in the See also:definitions of the signs > and < for the particular criterion in question. The See also:consideration of a See also:succession of events in See also:time will help to show that the assumptions made are not self-contradictory. An aggregate arranged in order by a definite criterion will be called an ordered aggregate. Let a, b be any two elements of an ordered aggregate, and suppose a<b. All the elements c (if any) such that a<c<b are said to fall within the See also:interval (a, b). If an element b, posterior to a, can be found so that no element falls within the interval (a, b), then a is said to be isolated from all subsequent elements, and b is said to be the element next after a. So if b'<a, and no element falls within the interval (b', a), then a is isolated from all preceding elements, and b' is the element next before a. As will be seen presently, for any assigned element a, either, neither, or both of these cases may occur. An aggregate A is said to be well-ordered (or normally ordered) when, in addition to being ordered, it has the following properties: (1) A has a first or lowest element a which is anterior to all the See also:rest; (2) if B is any part of A, then B has a first element. It follows from this that every part of a well-ordered aggregate is itself well-ordered. A well-ordered aggregate may or may not have a last element. Two ordered aggregates A, B are said to be similar (AP-LB) when a one-one correspondence can be set up between their elements in such a way that if b, b' are the elements of B which correspond to any two elements a, a' of A, then b> b' or b< b' according as a > a' or a < a'. For example, (1, 3, 5, ... )" (2, 4, 6, ... ), because we can make the even number an correspond to the See also:odd number (2n—1) and conversely. Similar ordered aggregates are said to have the same order-type. Any definite order-type is said to be the ordinal number of every aggregate arranged according to that type. This somewhat vague See also:definition will become clearer as we proceed. 4. The Natural See also:Scale.—Let a be any element of a well-ordered aggregate A. Then all the elements posterior to a form an aggregate A', which is a part of A and, by definition, has a first element a'. This element a' is different from a, and immediately succeeds it in the order of A. (It may happen, of course, that a' does not exist; in this case a is the last element of A.) Thus in a well-ordered aggregate every element except the last (if there be a last element) is succeeded by a definite next element. The ingenuity of See also:man has See also:developed a symbolism by means of which every See also:symbol is associated with a definite next succeeding symbol, and in this way we have a set of visible or audible signs r, 2, 3, &c. (or their verbal equivalents), representing an aggregate in which (r) there is a definite order, (2) there is a first term, (3) each term has one next following, and consequently there is no last term. Counting a set of objects means associating them in order 'with the first and subsequent members of this conventional aggregate. The process of counting may lead to three different results: (r) the set of objects may be finite in number, so that they are associated with a part of the conventional aggregate which has a last term; (2) the set of objects may have the same power as the conventional aggregate; (3) the set of objects may have a higher power than the conventional aggregate. Examples of (2) and (3) will be found further on. The order-type of i, 2, 3, &c., and of similar aggregates will be denoted by w; this is the first and simplest member of a set of transfinite ordinal numbers to be considered later on. Any finite number such as 3 is used ordinally as representing the order-type of r, 2, 3 or any similar aggregate, and cardinally as representing the power of r, 2, 3 or any equivalent aggregate. For reasons that will appear, w is only used in an ordinal sense. The aggregate r, 2, 3, &c., in any of its written or spoken forms, may be called the natural scale, and denoted by N. It has already been shown that N is infinite: this appears in a more elementary way from the fact that (r, 2, 3, 4,. . • )` (2, 3, 4, 5,. . . ), where each element of N is made to correspond with the next following. Any aggregate which is equivalent to the natural scale or a part thereof is said to be countable. 5. Arithmetical Operations.—When the natural scale N has once been obtained it is comparatively easy, although it requires a long process of See also:induction, to define the arithmetical operations of addition, multiplication and involution, as applied to natural numbers. It can be proved that these operations are See also:free from See also:ambiguity and obey certain formal See also:laws of See also:commutation, &c., which will not be discussed here. Each of the three See also:direct operations leads to an inverse problem which cannot be solved except under certain implied conditions. Let a, b denote any two assigned natural numbers: then it is required to find natural numbers, x, y, z such that a=b+x, a=by, a=zb respectively. The solutions, when they exist, are perfectly definite, and may be denoted by a—b, a/b and 4j a; but they are only possible in the first case when a>b, in the second when a is a multiple of b, and in the third when a is a perfect bth power. It is found to be possible, by the construction of certain elements, called respectively negative, fractional and irrational numbers, and zero, to remove all these restrictions. 6. There are certain properties, See also:common to the aggregates with which we have next to See also:deal, analogous to those possessed by the natural scale, and consequently justifying us in applying the term number to any one of their elements. They are stated here, once for all, to avoid repetition; the verification, in each case, will be, for the most part, left to the reader. Each of the aggregates in question (A, suppose) is an ordered aggregate. If a, /3 are any two elements of A, they may be combined by two definite operations, represented by + and X, so as to produce two definite elements of A represented by a+/3 and aX(3 (or 0); these operations obey the formal laws satisfied by those of addition and multiplication. The aggregate A contains one (and only one) element t, such that if a is any element of Aalso t<2c<3c . . . We may See also:express this by saying that A contains an See also:image of the natural scale. The element denoted by c may be called the ground element of A. 7. Negative Numbers.—Let any two natural numbers a, b be selected in a definite order a, b (to be distinguished from b, a, in which the order is reversed). In this.way we obtain from N an aggregate of symbols (a, b) which we shall See also:call couples, or more precisely, if necessary, polar couples. This new aggregate may be arranged in order by means of the following rules:—Two couples (a, b), (a', b') are said to be equal if a+b'=a'+b. In other words (a, b), (a', b') are then taken to be equivalent symbols for the same thing. If a+b'>a'+b, we write (a, b)>(a', b'); and if a+b'<a'+b, we write (a, b) <(a', b'). The rules for the addition and multiplication of couples are: (a, b)+(a', b')=(a+a', b+b') (a, b) X (a', b') = (aa'+bb', ab' +a'b). The aggregate thus defined will be denoted by N; it may be called the scale of relative integers. If c denotes (2, r) or any equivalent couple, (a, b)+e= (a+2, b+r)> (a, b) and (a, b) Xc= (2a+b, a+2b) = (a, b). Hence e is the ground element of Sr. By definition, 2t=c+c=(4, 2) = (3, r): and hence by induction mt=(m+r, r), where m is any natural integer. Conversely every couple (a, b) in which a>b can be expressed by the symbol (a—b)c. In the same way, every couple (a, b) in which b>a can be expressed in the form (b—a)i , where t' = (I, 2). 8. It follows as a formal consequence of the definitions that L+i = (2, r)+(r, 2) = (3, 3) = (r, r). It is convenient to denote (r, r) and its equivalent symbols by o, because (a, b) +(I, r)=(a+r, b+r) =(a,b) (a, b)X(r, r)=(a+b, a+b)=(r, r); hence c+i = o, and we can represent N by the See also:scheme—... 31", 2t', L', 0, c, 2t, 3L .. . in which each element is obtained from the next before it by the addition of t. With this notation the rules of operation may be written (m, ii, denoting natural numbers) mr+nt =(m+n)c mt'+nt' =(m+n)i m4+nc' = (m —n)L if m > n =(n—m)L' ,n<n m&Xnc=mnc, m~ Xnt'=mnc, xnt' -- nine, with the See also:special rules for zero, that if a is any element of N, a+O=a, aXO=o. To each element, a, of S I' a definite element a' such that a+a'=o; if a=o, then a'=o, but in every other case a, a' are different and may be denoted by mt, me'. The natural number m is called the See also:absolute value of me and mt.'. q. If a, 3 are any two elements of N, the See also:equation E+13=a is satisfied by putting =a+(3'. Thus the symbol a—f3 is always interpretable as a+13', and we may say that within TN- subtraction is always possible; it is easily proved to be also free from ambiguity. On the other hand, a/0 is intelligible only if the absolute value of a is a multiple of the absolute value of O. The aggregate N- has no first element and no last element. At the same time it is countable, as we see, for instance, by associating the elements o, at, bt' with the natural numbers r, 2a, 2b+1 respectively, thus (N)r,2,3,4,5,6,... (N) o, t, t', 2c, 2i , 3c.. . It is usual to write +a (or simply a) for at and —a for at'; that this should be possible without leading to confusion or ambiguity is certainly remarkable. ro. Fractional Numbers.—We will now derive from N a different aggregate of couples [a, b] subject to the following rules: The symbols [a, b], [a', b'], are equivalent if ab' = a'b. According as ab' is greater or less than a'b we regard [a, b] as being greater or less than [a', b']. The formalae for addition and multiplication (e included), then a+e> a, and cu= a. Thus A contains the are [a, bl +[a', b'] = [ab'-J-a'b, bb'] elements c, t+t, c+t+c, &c., or, as we may write them, t, 2c, [a, b] X [a', b'] = [aa', bb']. 3L, . . . me . . . such that me+nc=(m+n)t and mmXnc =mnc; All the couples [a, a] are equivalent to [r, r], and if we denote this by v we have [a, b]-Fu= [a+b, b]> [a, b], [a, b] X v = [a, b], so that u is the ground element of the new aggregate. Again 2v=v+v=(2, I), and by induction mu=[m, r]. More-over, if a is a multiple of b, say mb, we may denote [a, b] by ma. I r. The new aggregate of couples will be denoted by R. It differs from N and N in one very important respect, namely, that when its elements are arranged in order of magnitude (that is to say, by the See also:rule above given) they are not isolated from each other. In fact if [a, b]=a, and [a', b']=a', the element [a+a', b+b'] lies between a and a'; hence it follows that between any two different elements of R we can find as many other elements as we please. This See also:property is expressed by saying that R is in close order when its elements are arranged in order of magnitude. See also:Strange as it appears at first sight, R is a See also:count-able aggregate; a theorem first proved by G. Cantor. To see this, observe that every element of R may be represented by a " reduced " couple [a, b], in which a, b are See also:prime to each other. If [a, b], [c, d] are any two reduced couples, we will agree that [a, b] is anterior to [c, d] if either (I) a+b< c+d, or (2) a+b= c+d, but a<c. This gives a new criterion by which all the elements of R can be arranged in the succession [I, I], [I, 2], [2, r], [I, 3], [3, I,], [I, 4], [2, 3], [3, 2], [4, I]. which is similar to the natural scale. The aggregate R, arranged in order of magnitude, agrees with N in having no least and no greatest element; for if a denotes any element [a, b], then [2a-I, 2b]<a, while [2a+I, 2b]>a. 12. The See also:division of one element of R by another is always possible; for by definition [c, d] X [ad, bcj = [acd, See also:bcd] = [a, b], and consequently [a, b]= [c, d] is always interpretable as [ad, bc]. As a particular case [m, I]= [n, 11= [m, n], so that every element of R is expressible in one of the forms mu, my/nv. It is usual to omit the symbol v altogether, and to represent the element [m, n] by See also:min, whether m is a multiple of n or not. Moreover, m/r is written m, which may be done without confusion, because m/r+n/r=(m+n)/I, and m/IXn/I=mn/r, by the rules given above. 13. Within the aggregate R subtraction is not always practicable; but this See also:limitation may be removed by constructing an aggregate R related to R in the same way as N to N. This may be done in two ways which lead to equivalent results. We may either form symbols of the type (a, 0), where a, (3 denote elements of R, and apply the rules of § 7; or else form symbols of the type [a, 0], where a, 13 denote elements of N, and apply the rules of §ro. The final result is that R contains a zero element, o, a ground element v, an element v' such that v+v' = o, and a set of elements representable by the symbols (m/n)v, (m/n)v'. In this notation the rules of operation are mu+nz'u= (mn'+m'n) m , m' ,_ (mn'+m'n\ ,; n n' \ nn u n v +n' nn' Ju m m' ,mn'-m'n m'n-mn' = , n n, nn v, or nn' v', as See also:inn >or<m n; m m' mm' m, m', m m', mm',; n uXn' u=nn, u=nu Xn, v —vXn' u =--v' n, u m m' mn' m,, m', m m', mn', rn,.m' nu_n' u=m,nu=nu -n, nu_n,u =m,nu =n-u ---v; a-/i=a+fl', where 13+/3'=o; a+0=a, aXo=o. Here a and denote any two elements of R. If (3= (m/n)v, then /3' _ (m/n)v', and if (3 = (min) v', then /3' = (m/n)v. If (3 = 0, then (3'=o. _ _ 14. When R is constructed by means of couples taken from N, we must put [me, tit] = [mt', nL'] = (m/n)v, [mu, n,'] = [mL', (m/n)v', and [o, a]=o, if a is any element of N except o. The symbols [o, o] and [a, o] are inadmissible; the first because it satisfies the definition of equality (§ to) with every symbol [a, (3], and is therefore indeterminate; the second because, according to the rule of addition, [a, o] +[L. I] _ [ai., o] _ [a, o], which is inconsistent with +u> . In the same way, if o denotes the zero element of R, and any other element, the symbol o/o is indeterminate, and l;/o in-admissible, because, by the formal rules of operation, /o+v = /o, which conflicts with the definition of the ground element v. It is usual to write +n m (or simply n) for n v, and -n for mv'. Each of these elements is said to have the absolute value m/n. The criterion for arranging the elements of R in order of magni-
tude is that, if , .q are any two elements of it, >q when t-ti is See also:positive; that is to say, when it can be expressed in the form (m/n)v. -
15. The aggregate R is very important, because it is the simplest type of a See also: Every separation R=A+A' which satisfies these conditions is called a cut (or See also:section), and will be denoted by (A, A'). We have seen that every rational number a can be associated with a cut. Conversely, every cut (A, A) in which A has a last element a is perfectly definite, and specifies a without ambiguity. But there are other cuts in which A has no last element. For instance, all the elements (a) of R such that either a o, or else a> o and See also:a2 <2, form an aggregate A, while those for which a> o and a2> 2, form the complementary aggregate A'. This separation is a cut in which A has no last element; because if p/q is any positive element of A, the element (3p+4q)/(2p+3q) exceeds p/q, and also belongs to A. Every cut of this See also:kind is said to define an irrational number. The See also:justification of this is contained in the following propositions: (I) A cut is a definite concept, and the assemblage of cuts is an aggregate according to definition; the generic quality of the aggregate being the separation of R into two complementary parts, without altering the order of its elements. (2) The aggregate of cuts may be arranged in order by the rule that (A, A') < (B, B') if A is a part of B. (3) This criterion of arrangement preserves the order of magnitude of all rational numbers. (4) Cuts may be combined according to the laws of algebra, and, when the cuts so combined are all rational, the results are in agreement with those derived from the rational theory. As a partial See also:illustration of proposition (4) let (A, A'), (B, B') be any two cuts; and let C' be the aggregate whose elements are obtained by forming all the values of a'+/3', where a' is any element of A' and /3' is any element of B'. Then if C is the See also:complement of C', it can be proved that (C, C') is a cut; this is said to be the sum of (A, A') and (B, B'). The difference, product and quotient of two cuts may be defined in a similar way. If n denotes the irrational cut chosen above for purposes of illustration, we shall have n2 = (C, C') where C' comprises all the numbers a'/3' obtained by multiplying any two elements, a', /3' which are rational and positive, and such that a 2> 2, 13'2> 2. Since a'2/3'2> 4 it follows that a'/3' is positive and greater than 2; it can be proved conversely that every rational number which is greater than 2 can be expressed in the form a'/3'. Hence n2=2, so that the cut n actually gives a real arithmetical meaning to the positive root of the equation x2 = 2 ; in other words we may say that n defines the irrational number d 2. The theory of cuts, in fact, provides a logical basis for the treatment of all finite numerical irrationalities, and enables us to justify all arithmetical operations involving the use of such quantities. 17. Since the aggregate of cuts (ZT say) has an order of magnitude, we may construct cuts in this aggregate. Thus if a is any element of ZT, and £4 is the aggregate which consists of a and all anterior elements of U, we may write n=a+ a', and (a, a') is a cut in which a has a last element a. It is a remarkable fact that no other kind of cut in U is possible; in other words, every conceivable cut in 27 is defined by one of its own elements. This is expressed by saying that ZT is a continuous aggregate, and ZT itself is referred to as the numerical continuum of real numbers. The property of continuity must be carefully distinguished from that of close order (§ II); a continuous aggregate is necessarily in close order, but the converse is not always true. The aggregate IT is not countable. 18. Another way of treating irrationals is by means of sequences. A sequence is an unlimited succession of rational numbers al, as, as . . . am, a",+1 .. . (in order-type w) the elements of which can be assigned by a definite rule, such that when any rational number e, however small, has been fixed, it is possible to find an integer m, so that for all positive integral values of n the absolute value of (am+n—a,,) is less than e. Under these conditions the sequence may be taken to represent a definite number, which is, in fact, the limit of a", when m increases without limit. Every rational number a can be expressed as a sequence in the form (a, a, a, ...), but this is only one of an infinite', variety of such representations, for instance = (.9, •99, .999, . . .) = I 4' 2I . ~2'$,. 2n and so on. The essential thing is that we have a mode of re-presentation which can be applied to rational and irrational numbers alike, and provides a very convenient symbolism to express the results of arithmetical operations. Thus the rules for the sum and product of two sequences are given by the formulae (al, as, a3, . . .) + (bl, b2, b3, . . .) _ (a,+bl, a2+b2, as+ba . . . ) (al, as, a3, . . .) X (bl, b, b3, . . .) _ (albs, a2b2, a3b3 . . . ) from which the rules for subtraction and division may be at once inferred. It has been proved that the method of sequences is ultimately equivalent to that of cuts. The See also:advantage of the former lies in its convenient notation, that of the latter in giving a clear definition of an irrational number without having recourse to the notion of a limit. 19. Complex Numbers.—If a is an assigned number, rational or irrational, and n a natural number, it can be proved that there is a real number satisfying the equation xn=a, except when n is even and a is negative: in this case the equation is not satisfied by any real number whatever. To remove the difficulty we construct an aggregate of polar couples {x, y}, where x, y are any two real numbers, and define the addition and multiplication of such couples by the rules {x, yl+(x', y'}={x+x', y ;-y i; (x, yl X x', y'l = xx'—yy', xy'+x'y}. We also agree that {x, y} < {x', y'}, if x<x' or if x=x' and y<y'. It follows that the aggregate has the ground element {I, o},which we may denote by v; and that, if we writer for the element {o, i }, 42=t—I, o}=—a. Whenever m, n are rational, {m, n} =mo+nr, and we are thus justified in See also:writing, if we like, xa+yr for {x, y} in all circumstances. A further simplification is gained by writing x4nstead of xa, and regarding r as a symbol which is such that See also:r2= -1, but in other respects obeys the ordinary laws of operation. It is usual to write i instead of r; we thus have an aggregate 3 of complex numbers x+yi. In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by to, o}, the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by See also:Gauss and See also:Cauchy), ifao, al, . . . an are any assigned real or complex numbers, the equation clean .+alz"-1+... +an-lz+an =o, is always satisfied by precisely n real or complex values of z, with a proper See also:convention as to multiple roots. Thus any algebraic See also:function of any finite number of elements of 3 is also contained in 3, which is, in this sense, a closed arithmetical field, just as ZZ is when we restrict ourselves to rational operations. The power of 3 is the same as that of n. 2o. Transfinite Numbers.—The theory of these numbers is quite recent, and mainly due to G. Cantor. The simplest of them, w, has been already defined (§ 4) as the order-type of the natural scale. Now there is no logical difficulty in constructing a scheme u1, u2, u3 . . . (v1, indicating a well-ordered aggregate of type w immediately followed by a distinct element vl : for example, we may think of all positive odd integers arranged in ascending order of magnitude and then think of the even number 2. A scheme of this kind is said to be of order-type (w+1); and it will be convenient to speak of (w+ I) as the See also:index of the scheme. Similarly we may form arrangements corresponding to the indices w+2, w+3 . . . w+n, where n is any positive integer. The scheme ul, u2, n3 . . . I vl, V2, V3 . Additional information and CommentsThere are no comments yet for this article.
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