Arithmetical Composition and Inversion of Functions Over Classes
Eric Temple Bell · Transactions of the American Mathematical Society · 1931
A class will be called countable if it contains either only a finite number or a denumerable infinity of distinct elements.4. Abstract equality, indicated by =, has its usual significance as a reflexive, symmetric, transitive relation, with the postulate (or theorem, as the case may be in a given context) that, if K |x, y, where K is any non-null class, and if x = y, then either of x, y may replace the other in any relation concerning x or y or both.The relation ~ of §21 is an instance of =, and is such, not by postulation, but by proof.All classes in the sequel are postulated to be such that equality is significant for the elements of any given class.The sign .d . of implication will be used where convenient.5. Capital Greek letters shall denote operations.K is any non-null class, t is an arbitrary constant integer >0.A capital Latin superscript Ot, Ct, Pt, 0, C, P, B, A, D, ■ ■ ■ , as in $BüT, connotes a property of 0tK: asserts that the result (xi, • • ■ , x,)* of operating with $ on the one-row matrix (xi, • • • , x¡) is uniquely known whenever K \xx, • • ■ ,xt and (xi, • ■ • , X() is given.If i = l, by convention* (x)*=x* = x whenever K \x.$CtK: asserts 0; QCK: asserts $CtK for all finite integers i>0; 0; $BK : asserts $CK and that (Xi, • • • , Xr, Xr+i)* = ((Xi, ■ • • , Xr)*, Xr+i)* for all integers r>0, whenever Xi, • • • , xr, xr+i are in K and (xi, • • • , xr, xr+i)*, (xi, • • • , xr)* are defined.By convention, if <&BK, then (x) =x*=x whenever K |x.This convention as a separate statement is unnecessary, by the previous convention, but is included on account of its importance.* The restriction on * imposed by this convention is only apparent, and is a mere convenience of notation.The important case where the convention need not hold is provided for in the discussion of functions in §21, for which a different notation is used.