Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/525

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errata : p. 503, line 14, for "*88.38" read "*88.36."

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analogous to the infinite cardinal product. This will be explained at a later stage (*172). Although it is chiefly €A’" and FA’" that will be required in the sequel, we shall treat P A’"generally, because this introduces little extra com- plication, and most of the theorems which hold for €A’" or FA’" have exact analogues for P A’ K, P A’K, as above defined, is the class of one-many relations contained in P and having " for their converse domain, We know of no proof that there always are such relations when" C G’ P. In fact, the proposition "C G’P. P,IC. a! PA’K is equivalent to the" multiplicative axiom," i.e. to the axiom that, given any class of mutually exclusive classes, none of which is null, there is at least one class formed of one member from each of these classes. (This equivalence is proved in *88.38, below.) It is also equivalent to Zermelo’s axiom *, which is ( ex) . a ! €A’ 01 ex ’ex ; hence also it is equivalent to the proposition that every class can be well- ordered. In the absence of evidence as to the. truth or falsehood of these various propositions, we shall not assume their truth, but shall explicitly introduce them as hypotheses wherever they are relevant. In the present section, we shall begin (*80) by considering such properties of PA’K as do not depend upon any hypothesis as to P. We shall then (*81) proceed to consider such further properties of PA’K as result from the hypothesis PI K i: CIs -+ 1, This hypothesis is important, because it is verified in many of the applications we wish to make, and because it leads to important properties of P A’K which are not true in general when J is not subject to any hypothesis. These special properties are mostly due to the fact that when PI" is a many-one relation, 1) A’" consists of one-one relations (not merely of one-many relations, as it does in the general case). This is proved in *81.1, We then (*82) proceed to consider the case of relative products, i.e. (P I Q)A ’x. It will appear that, with a suitable hypothesis, (PIQ)A’X=IQ"PA’Q"A. and D"(PIQ)A’X=D"PA’Q"X. In the following number (*83) we apply the results of *80 to the particular case where P is replaced by €, which is the important case for cardinal arithmetic. In *84 we apply the propositions of *81 to the case where P is replaced by €, and where, therefore, we have the hypothesis € I K € CIs -+ 1. This hypothesis is equivalent to the hypothesis that no two members of K have any members in common, i.e. that ex, {3 € K . ex +: {3 · , fJ . ex n {3 = A. [1]

  1. * See his "Beweis, dass jede Menge wohlgeordnet werden kann," Math. Annalen, Vol. LIK, pp. 514-516.