 One of Skolem’s stated goals was to present a logical foundation for number theory which avoids the use of unrestricted quantifiers. He was inspired in this regard by the observation that it is possible to develop much of elementary arithmetic without the use of the expressions “always” (i.e. for all) and “sometimes”  (i.e. there exists) which figure in the formalization of number theory given by Russell and Whitehead in Principia Mathematica (1910–1913).  This was to be accomplished by formulating arithmetical theorems as what he referred to as functional assertions.  These took the form of identities between terms defined by primitive recursive operations which Skolem referred to as descriptive functions.  For instance, the commutativity of addition is expressed in this form by an equation with free variables 