50 pitcher: interrelations of 



The propositions of § 44.1, 5; § 42.7, 812; § 41.1, 2, 6, 7, 12, 14 express 

 relations among the properties L; C; D; A; A; Ki; K,; ivj,^ as prop- 

 erties of classes of functions on ^^ finite; 'ip dual; '^ singular. The 

 effect of these propositions is to cut down the number of existent 

 composite properties to 88 for '^ finite, to 82 for "i}.^ dual, and to 20 for 

 ^ singular. 



The reader will have no difficulty in verifying the table by means 

 of the propositions cited above, and the lists of examples given in the 

 sequel. The examples given are all examples of classes of functions 

 on "ip"', "ip"', ^1?"-, or '^^ so that of the 145 composite properties proved 

 to be existent there is no loss in passing from '^ unrestricted to '^ 

 denumerable, nor in passing from ^p"" to ^^'^K The 88 existent com- 

 posite properties on ^^'■^ include all existent non-linear composite 

 properties. 



Theorem I. Of the 2* = 256 propositions involved in the complete 

 existential theory of the properties : 



L; C; D; A; A; A',; A'^; A12*, 

 of systems : 



(21; l^; A; m, 



145 are propositions of existence and 108 are propositions of non- 

 existence* Of the properties : 



L; C; D; A; A; Ai; A'o, 



any six are completely independent. In addition the properties in each 

 of the sets of five : 



(C; D; A; A; A„J, (L; D; A; A; A„J, (L; C; D; A; A,,J, 



are completely independent. Any other subset of properties completely 

 independent is a subset of one of the sets already declared to be com- 

 pletely independent. 



The proof of the complete independence in the various cases above 

 consists in \-erif3'ing that the appropriate examples are given in the 

 lists of the sequel. The proof of the last statement of the theorem 

 consists in showing by means of the propositions referred to above 

 that no subset of seven properties is completely independent, that no 

 other subset of six or five properties is completely independent, and 



* The three composite properties about which we have not decided are (L C~D A A 

 K,K^-K,,^); (L C-D-A A K.KrK,,^); (L-C-D A A K,K,-K,,^). 



