BY A. N. WHITEHEAD,Relbw of Trinity College, Cambridge, England

BY A. N. WHITEHEAD,Relbw of Trinity College, Cambridge, England.

The following memoir is concerned solely with the theory of Cardinal numbers, and does not touch upon the theory of ordered aggregates or of ordinal types which form the other parts of the general theory of aggregates (mengenlehre).* Sections I and I1 are preliminary, to explain the notation and methods of reasoning. Incidentally they may be of independent service as explaining the elements of Peano's developments of mathematical logic? (Section I) and of Russell's symbolism for the Logic of Relations1 (Section 11). I believe that these two methods are almost indispensable for the development of the theory of Cardinal numbers. The abstract nature of the subject makes ordinary language totally ineffective, only gaining precision by verbosity, and imagination is very mislearling, since it presents to us special aggregates which are denumerable or of the power of the continuum. Thus we are thrown back onto a strict logical deduction by a, symbolical method. I believe t,hat the invention of' the Peano and Russell symbolism, used here, forms an epoch in mathenlatical reasoning.

*Due almost entirely to G.Cantor. Almost all that is known respecting Cardinal Numbers, apart from their relations to the other parts of the theory, is given in his memoir in Math. Annal., Bd. XLVI, 1895 : this and its sequel on Ordinal Types, loc. cit. Bd. XLIX, can be obtained as one pamphlet in a French translation, translated by F. Marotte, Paris, A. Hermann, 1899. For a general sketch of the whole theory, with bibliography, cf. "Die Entwickelung der Lehre von den Punli-tmannigfaltigkeiten " : Bericht von A. S~honflies,~'Yahresberichtder Deutschen Xathernatiker.Vereinigung,"1900. For a development of the theory of Ordinal Types in respect to condensed series, by the methods of the present memoir, cf. Bertrand Rassell, Revue de Mathhmatiques, vol. VII, Turin, 1901, where the methods of reasoning used here were first elaborated. t For references to Peano's articles, cf. my memoir on the " Algebra of Symbolic Logic," in this Journal, vol. XXIII. $ Cf. Russell, loc. cit. 48 368 WHITEHEAD: On Cardinal Numbers.

Section I11 is entirely due to Russell, and is written by him throughout. It deals with the principles of the subject, and effects a considerable simplifica- tion in them. Two classes are defined [cf. * 1-1 (all references are to the series of propositions commencing with Section I11 and thence numbered consecu- tively)] as similar when a one-one relation can be established between their numbers. The cardinal number (or power) of a class u is defined simply as the class of those classes similar to ts (cf. * 1-3); thus, for instance, the cardinal number 2 is the class of those classes which (in ordinary language) contain two members, and we shall say that a class containing two members is a 2. The finite cardinals are defined [cf. * 1.4 to * 1.6 11 as those cardinal numbers which can be obtained by induction.* The class of infinite cardinals is defined [cf.* 1.8) as the class of cardinals which are not finite. The whole class of cardinal numbers is denoted by Nc. The properties of finite and infinite cardinals are deduced by strict logical reasoning from these definitions. In particular, if the cardinal number of the class of finite cardinals is denoted by a,, it is proved that every infinite class contains a class whose cardinal number is a, [cf: * 2-61, and that no finite class is similar to any part of itself [cf. * 2-75], and that every infinite class is similar to some part of itself [cf. * 2-76], The cardinal and ordinal theo- ries of finite number are connected in 3.3. The known properties of a. are proved [*4.1 to * 4-24]. The theorems that if cr, and ,6 are any two cardinals, and a is infinite and a 2 P, then a+p=a and a~p=a: cannot be deduced from these premises, and, as far as I am aware, have never been proved. Russell points out that the first of them follows from another unproved theorem [*4*3]. The class of numbers for which the first theorem holds is considered (cf. * 30). The second of the theorems cannot be thus deduced; but in Section V, * 20.1, it is proved that if a belong to the class for which the first theorem holds and P 2 8, then

The similar cases for a + P =a are examined in * 30.0 to * 31.3. Russell also gives exact proofs of such theorems as, /3

* The recognition of the importance of Induction is due to Dedekind, cf. "Was sind and was sollen die Zahlen ?" 1887. WHITEHEAD: On Cardinal Numbers. 369 a + PIa- +y, and axPSa x y, and ~~FGcY,and paLya [cf. *4*37, k4.39, *4.4, *4-41]. Also *4.5 contains another proof (assuming 4-3) of Bernstein's and Schroder's theorem, that a 2P and p 2a imply a =P. Throughout Sections IV and V, I am indebted to Russell for great improve- ments in the symholism of the proofs and for the correction of many oversights. Section IV is devoted to the general consideration of addition and multiplication when the number of surrlxnands or of factors is not necessarily finite or even denumerable. Cantor's definitions only apply to these cases [cf. 7.0 to * 7.211. The associative laws are proved [cf. *8-1 to *8*31, and * 10.0 to * 10*22] both for addition and xnultiplication when the number of brack- ets, as well as the number of numbers in each bracket, is not necessarily finite or denumerable. With these definitions the co~nmutativelaws become entirely irrelevant. The ordinary connection between addition and multiplication, exemplified in a + a = 2 x a, is proved [cf. * 7.31 to hold gener- ally ; also the distributive law [cf. * 11-01. These definitions and theorems require several preliminary definitions; thus for instance [*5.11, that of a "class of classes exclusive," and [*6.01 of the "multiplicative class" of a class of classes exclusive ; this last conception assumes great importance in the sequel ; also, the general definitions of sums and products depend on the definition [*8.221 of " a class of classes of classes arithmetical." The remainder of Section IV is devoted to theorems following from these definitions. Section V opens with the definition [*12.11 of a power (pa); this definition follows from that of a product; and [*14.11 Cftntor's definition is deduced from it. Thus, by this procedure, the ordinary relations between addition, multiplication, and powers of numbers are preserved in their full generality. The theory of combinations is then extended to infinite cardinals; thus, if a, /3 are numbers, and a 2-,@,C; is defined [cf. * 15.1 and * 16*0]: the extension of Vandermonde's theorem is proved [cf. * 16.11 ; if a is infinite and such that a +a =a, 0;= 2" [cf. * 16-23] : extensions of the binomial theorem are given in * 17.0 to * 19*7; it is proved [cf, * 17-41 that if a and ,kl are any cardinals,

If a is infinite and v finite [cf. * 20-41, then va = 2", and when a is infinite and such that a2=a, then [cf. * 32-41, if p 5 2", we have Pa= 2". WHITEHEAD: On &rdinaZ Numbers. 381

*3*2 a, ,6~Nc.a=a,. [*2.6 .3 .Prop] . '2 ao=ao+ao, [R1=lReln P3 {xPy.=.3Ncfin n n3(xR2"y)/. p~=~'O.pl=~'l.).po~pl=A.po~p,=p.p~,plea,.).Prop]. *21 n~Ncfin .).ao=ao~.n, rJr4-2. Induct. ). Prop]. '22 a,= aoXao, [M=x3 {RNcfin n (m, ?z)~(x=I~u ~n)~.).~iM=a,~a~, (1) R'~l+l.p'=M,*.im~~nR'lrn'u~n'.=:m>n.rn-n>1. ml+nl=m+n.nt'=m-l.u.m>n.m-all. - m' + n' = m +n + 1 .n1= 0 :.) .p' E aO.).Prop]. -23 n~ Nc fin .] .ao=alf., [Induct] . xau.yrv 024 u,VE clsY~cfin.]*~;tX av=ao. 03 (u E cls infin. ].3cls'a, excl n v 3 (U 'v =u). PP. For definition of clsyaoexcl, cf * 5.1 later.) Note: It has been shown that every infinite class contains classes whose number is a,. Thus if a class v of aOJshas been found which contains all but a finite number of u's, these may be added on to the v7s; but if an infinite number of u's remain, new aO7smay be found out of them, and to this process there can 382 WHITEHEAD: On Cardinal Numbers.

be no end as long as any u's remain. It appears not impossible that a theorem may be discovered proving that this process can always be completed. In any case, it is much to be desired that the possibility in all cases of bhis comple- tion of the above process should be either proved or disproved. In order to show the importance of this investigation, the theorem is here hypothetically assumed and stated as a primitive proposition (*4*3), and some immediate deductions (*4.31 to *4-51) are made from it. The subject is ,resumed in the sets of propositions * 30, * 31, and * 32.

[A 4.3 .).Prop]. [a=aoP= (ao+ao)P= a +a]. 033 aao= a, [a = aoP=at@=aao]. '34 aeNc nP3 (a= aoP), [*4.33. ).Prop] . 035 a>ao.Pao.3.NcnP3(a=aOP) =la, [*4-35 .).Prop]. -37 P.~Uv)u~ w.').Prop]. 038 P.ap<=ay, [ve~.wey.v)w.w)a.). uCv)ucw.).Prop]. 04 p

[This theorem is enunciated and proved in Section V, * 20.1 ; it is men- tioned here for the sake of completeness.]

Note : This is Schroder's and Bernstein's theorem ; cf. Borel, "Thborie des Bunctions," Paris, 1898, p. 108 ff. [XL:~+I.C=~.Z~V.S~E~+I.G~=V.Z']U.>.~(~~)=~.~~(~)>~~/)~~,(I) (1) .pu = a .pv = p. pu-ifl = y .p5f-51(5) = 8 .). a~P.Pla.a=~+y.p=~+S.].a=a+~+8, (2) (2). *4*38.).y+8~a.).S~a.].af8=a.).P=a.).Prop]. Note : Bernstein's proof of Schroder's and Bernateinls theorem given by Borel (loc. cit.) does not depend upon *4.3. '51 ~,,~E.XC.~>P.).-(@La), [*4*5 .I.Prop],

SECTIONIT. *5 '1 clsaexcl.=.~l~2nd3{p,p1~d.].q=pf.LJ .pnql=~]. Df.

Note : excl is contracted from (( exclusive ". 02 a E c1s2. ).cls'a excl = cls'a n clsZexcl. *6 d c1s2 excl .):. *O dX=clsnm3{p~d.)I,.pnrn~1:m)u'dj.

Note : dX is the " multiplicative " class of d, and can be read as ic d multiplicative ": it has no existence, according to this definition, unless d is a cls2excl. 011 p~d.x~p.d~=d-~p.m~~d~.rn=mluLX.). m&dX, [Hp.pl~d.pl-=p.].pl&d,.x - &pl.).rn npr=ml fipI.1. m npl&l. (1) Hp.).mnp= LX.).~n,pel. (2) (1).(2).):ptt~d.).pl1n WEI .. . . (3) m) v 'd, (4) (3) (4) 3 Prop] *I2 u) ucd:x, y~u.x~'y.)~,~.-Zd np3(x, y&p):).XdX nm3(w)m), [Hp.p~d.Rpnzc.3.p nuel, (1) dU=dnp3(p nu=A).~.d,~cIs~excl.).3d,X, PI (1).ml&d~.m=ml~21.):p&d.)p.pnm~l) PI rn) u 'd .. .. (4)) (3) .(4) .3 .m E dX.D.Prop] . *I3 m,&dX.]. -TdX n m13(mto'nt .ml) m), [m'o'm. dam.) .3d n p3(mt np=A) .) .m!-cdX.). Prop]. 50 WHITEHEAD: On Cardinal Numbers. 393

The sets of propositions with integral numbers 80 to 32 will be devoted to these investigations ; the proofs of some propositions practically proved by Rus- sell in Section I11 will be omitted. In these sets of propositions (30 to 32) it will be found that the only properties of an Nc infin used are that [cf. 4.1)

where cxo is the first cardinal number.

Nao=Ncn/33[3Ncn 63(6>0./3=ao~6)]. Df. Note: As long as all the numbers considered belong to Nao,the proposi- t8ionsin Section 111, k4.31 to 4.51, hold without modification. GLE~~~~/~E~~c.).'.u~~.).u$P=~:~~P.).~$P=@, ralp.).ala+P(a+- a$a.).a+p=a, (1) a6p.).3Ncn y3(a+ y=p).).a+P=a+a+~=a+ y=p, (2) (1).(2) .3 .Prop]. p, y&Nc.3Naona3[pialy] .).P+y=y, [yip+ yla +yly.3.Prop-j. Note : This theorem is important, and enables the enunciations of some theorems of * 21 and It 22 to be extended. PENc.a~Na,nz3(zLp)..) .a2a0xP, [azp.) .aoxa1,a0xp.) .aLaoxP]. p&Nc.yeNc~z3(p(yLa~~P)~).a~~/3=a~~y, [*30.3 .).Prop]. a, at&Nao .a

Ncexp=Ncna3[XNcinfinnp3{3Ncn y3(y>1.a=yB)/]. DK u, E CISinfin .).p cls' u E Nc exp, [*15.0 .I. Prop]. a E Nc exp.). a >a,. Nc exp .).Nuo, [~PENcexp .).y> 1.).yao&Ncexp.). yao>ao, (I).]. YB~ao~yP~yao~yS( ya0+@, y@~Ncexp.).p~Ncinfin.).ao+p=@, (2). (3). 3. ys=ao~yB.).Prop].