Two theorems on experimental logics

A generalization of a formal system is considered, in which the axioms of the formal system can be withdrawn or supplemented, as mechanical experimentation proceeds through time and the consequences of various combinations of assumptions are realized. The "theorems" of these experimental logics are taken to be those assertions possessing a proof which remains valid for all "sufficiently large time." Under very broad hypotheses on experimental logics, we obtain the following 0 two results: (1) There is a (n infinitude of) true but unprovable II sentence (s) ; (2) There is no mechanical procedure for uniformly finding any (of the infinitude of) true but unprovable II -sentence(s) . Our first result is analogous to Godel's First Incompleteness Theorem for formal systems (which it implies). Our second result differs sharply with that for formal systems, where a true but unprovable II--sentence is mechanically obtained in the course of Godel's proof. TWO THEOREMS ON EXPERIMENTAL LOGICS 1 by R. G. Jeroslow If one experiments through time with mechanical processes, and, on the basis of the outcome of these "computer experiments," one revises the axioms of a mathematical system, then the resulting time-dependent deductive system is called an