Abstract

I hold up my hand and I count five fingers. I take it on faith that the mapping from fingers onto numbers is recursive in the sense of the mathematician’s definition of the informal concept, “human calculability following a fixed routine.” I cannot prove the mapping is recursive—there is nothing to prove! Of course, mathematicians can prove many theorems about recursiveness, moving forward, so to speak, once the definition of the concept “recursive” has been isolated. Moving backwards is more difficult and this is as it should be: for how can one possibly hope to prove that a mathematical definition captures an informal concept? This paper is about just that one word, capture, or more precisely the rela- tion “x captures y,” and the question, if y is taken to be computability, definabil- ity or provability, does there exist an adequate and unique choice of x?