An Abstract Semantics for Atoms in Nuprl∗

With the standard inference rule set for Nuprl, the type Atom cannot be proved either to be finite or infinite, despite the fact that any character string (over a certain finite alphabet) can be used to form a canonical expression for a member of the type. For each k ∈ N one can prove in the logic a formula to the effect that there are at least k atoms. One can do this simply by exhibiting k distinct character strings as constants. But one cannot prove a statement to the effect that for all k ∈ N there are at least k atoms. Similarly, one cannot in the logic provide an enumeration of the Atoms, nor show that there are finitely many Atoms. This gap in provability may seem strange in a logic, though it is not so strange as a way of treating some data values as purely atomic. For example, the tokens that make up members of enumeration types in Wirth’s well-known language PASCAL cannot be analyzed either, as opposed to the atoms of some LISP dialects which have explode and implode operations that convert between atom values and character strings. A pragmatic reason for omitting such rules and operations from a logic might be to provide a convenient abuse-proof facility for treating a class of values as atomic. Our purpose here is to explain a principled semantic treatment of atoms that characterizes these expressive limits and actually invalidates the rules that would allow for demonstrating in the logic either finiteness or infiniteness of the atoms. Our approach here will be strictly semantic rather than ontic. An ontic approach might attempt to define a class of objects that are somehow impervious to reasoning by means of the rules we wish to invalidate. Perhaps one would stipulate some kind of ur-element or define some open-ended always finitely-extensible class of values. That is not the aim here. We will provide a “supervaluation” semantics that stipulates truth of assertions in the logic in terms of quantification over plenty of choices for interpreting the class of atoms and the strings denoting them. Note that this is not a matter of distinguishing