PROPERTIES OF CLASSES OF FUNCTIONS 6 



Analysis. Moore shows that the property A'lj^ is imphed by the 

 properties D; A; Ki; Ko and that the genus of systems: 



whose class W has the properties : 



D; A; A'l; A%, 



is closed under composition of classes and developments.! In §§33; 

 35; 40 we indicate three other extensive genera of systems: 



2 = {%;f;A;m), 



which are similarly closed under composition and whose classes 5D? 

 have this fundamental property A,,^. 



This paper had its origin in an attempt to develop the complete 

 existential theory, soon to be defined, of the properties: 



(2) L;C;D;A;A;A,;A,;A,,^. 



This study has proved to be extremely interesting and suggestive and 

 has given rise to many theorems not immediately connected with the 

 complete existential theory. Such an investigation in connection 

 with the fundamental postulates of other mathematical disciplines 

 would no doubt prove to be equally fruitful. In our investigations 

 we have made free use of the properties : 



(3) Z)i;Ai^;A2^, 



and many of the theorems involve these properties. 



A property P, defined for the elements of a certain class or for 

 systems of a certain type, is said to be existent or non-existent (for 

 that class or type) in case there exists or does not exist an element 

 of the class in question or a system of the type in question possessing 

 the property P. 



Consider in general n properties : 



• P 



of systems S of a certain type. Consider also the 2" composite 



* A system (3(; $; A; 3JI) is a class ^ of elements with a development A, and a class 3J! 

 of functions on *)} to 31. (21 denotes the class of all real numbers.) 

 1 1. G. A., § 84. 



