PROPERTIES OF CLASSES OF FUNCTIONS 17 



15. Some special developments. — The ordinary developments* of 

 ^1 = 1; ^>""^1, 2, ■•-,«; ^13'" = 1, 2, ■••,«, ••■; iV^'=mterval(0 1), 



respectively denoted by: 



A'; A"-; A"^; A'^ 



may be indicated as follows: 



A' 



A"" 

 A'" 



Aim.^. /„, = 1 . ^;^"'i = 1; 



A 9 m . D . /„, = n . f"' = I (l); 



A 9 7« . z) . /„, = m . 'ip™' = I (l); 



A J /« . z) . /„, = m + I . for every I the class 'iP'"' is the interval 



/ I -2 l\ 



\ m VI ) 



\ m-\-l 



where is taken as and is taken as 1. 



m m 



16. Properties of developments A derived from properties of stages 

 A". — Consider a property P of stages A" of a development A of a 

 class ^, in particular one of the three properties: 



(1) P = V (vacuous) ; F (fine); C (complete), 



which are about to be defined. 



A stage A" of the development A of the class '^ is said to be vacuous;, 

 fine; complete, respectively in case '$"' is null-class; no two elements 

 of 'i|3'" belong to the same class i^""'; '>]?"' = ^:i?. 



A property P of stages A™ gives rise to certain derived properties 

 of the development A. We consider three such derived properties, in 

 notation : 



(2) P; Pi; Po_. 



(3) A^: =:AaTO .3. A"'^, 



viz., the development A has the property P in case for every m the 

 stage A™ has the property P. 



(4) A'\ = . A 9 (3 mo 9 m > nio . z> . A""'), 



viz., the development A has the property Pi in case there exists an mo 

 such that for every 771 > mo the stage A"" has the property P. In 

 this case the development is said to possess the property P ultimately. 



*Cf. I. G. A., §§66a; 67a. 



