Aoo-Definable functions
S. W. P. Steen · Cambridge University Press eBooks · 1972
Calculable functions An A oo -function is an A oo -formula of type u, ui , …, these are called one-, two-, …place A oo -functions. If ρ is a one-place A oo -function and if v is a numeral then (ρ v ) is a numerical term and determines a unique numeral called the value of the function ρ for the argument v . Similarly if ρ is a two-place A oo -function and if v and K are numerals then ((ρ v ) K ) is a numerical term and so determines a unique numeral called the value of the function ρ for the arguments v and K , in that order. Similarly for many-place functions. A calculable function of natural numbers is a rule or set of rules such that given the ordered argument set we can by following the rules effectively find a natural number called the value of the function for that argument set, the value must be unique and a given function always requires the same number of arguments. Thus the A oo -functions are a particular kind of calculable function in that the rules for finding the value given the argument set is of a particular kind. For example a set of instructions to find the value of a function of natural numbers might be: replace (S(Sv)) by v as long as possible, this would replace an even number by 0 and an odd number by ( S O).