VARIABLE TRUTH

The logic of physical processes of computation has obvious temporal aspects since it must be able to describe facts (e.g., the contents of the registers of a computing machine) varying through time. Because of the loop phenomenon it also has an infinitary aspect. Thus the underling time scale of an appropriate temporal logic for Computations and their programs must have at least countably many infinite points, any two of which must be infinitely far apart. An initial segment of the transfinite ordinals containing countably many limit ordinals is therefore a good candidate for a computational scale of time. The systems of KROGER [7], KROGER [8], and of SZABO [13] are based on such a scale. However, recent investigations of RICHTER and SZABO [ I l l show that a more,satisfactory treatment can be given in terms of nonstandard models of arithmetic. It is the purpose of this paper to describe and axiomatizr the appropriate logic. A different temporal approach to computational logic is prrsrnted in KAWAI [5] and a literature on nonstandard programming theory is beginning to develop (cf. A N D R ~ K A , N~METI and SAIN [l], B I R ~ [Z], CZIRMAZ [3], and X ~ M E T I [9]).