Principia mathematica, vol. 1 (of 3)

[12] See the last section of the present Chapter. Cf. also H. Poincaré, "Les mathématiques et la logique," Revue de Métaphysique et de Morale, Mai 1906, p. 307.

[13] When the word "function" is used in the sequel, "propositional function" is always meant. Other functions will not be in question in the present Chapter.

[14] We shall speak in this Chapter of "values for " and of "values of ," meaning in each case the same thing, namely , , , etc. The distinction of phraseology serves to avoid ambiguity where several variables are concerned, especially when one of them is a function.

[15] We use "always" as meaning "in all cases," not "at all times." Similarly "sometimes" will mean "in some cases."

[16] See Chapter III.

[17] Note that statements concerning the significance of a phrase containing "" concern the symbol "," and therefore do not fall under the rule that the elimination of the functional ambiguity is necessary to significance. Significance is a property of signs. Cf. p. 43.

[18] Cf. Chapter III.

[19] When we speak of "values of " it is , not , that is to be assigned. This follows from the explanation in the note on p. 42. When the function itself is the variable, it is possible and simpler to write rather than , except in positions where it is necessary to emphasize that an argument must be supplied to secure significance.

[20] When a (finite) set of predicates is given by actual enumeration, their disjunction is a predicate, because no predicate occurs as apparent variable in the disjunction.

[21] Note that in this definition the second sign of equality is to be regarded as combining with "Df" to form one symbol; what is defined is the sign of equality not followed by the letters "Df."

[22] Here the combination or disjunction is supposed to be given intensionally. If given extensionally (i.e. by enumeration), no assumption is required; but in this case the number of predicates concerned must be finite.

[23] "Una questione sui numeri transfiniti," Rendiconti del circolo matematico di Palermo, Vol. XI. (1897). See *256.

[24] This contradiction was suggested to us by Mr G. G. Berry of the Bodleian Library.

[25] is the number of finite integers. See *123.

[26] Cf. König, "Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem," Math. Annalen, Vol. LXI. (1905); A. C. Dixon, "On 'well-ordered' aggregates," Proc. London Math. Soc. Series 2, Vol. IV. Part i. (1906); and E. W. Hobson, "On the Arithmetic Continuum," ibid. The solution offered in the last of these papers depends upon the variation of the "apparatus of definition," and is thus in outline in agreement with the solution adopted here. But it does not invalidate the statement in the text, if "definition" is given a constant meaning.

[27] Cf. Poincaré, "Les mathématiques et la logique," Revue de Métaphysique et de Morale, Mai 1906, especially sections VII. and IX.; also Peano, Revista de Mathematica, Vol. VIII. No. 5 (1906), p. 149 ff.

[28] The solution of Burali-Forti's paradox by means of the theory of types is given in detail in *256.