TY - JOUR
TI - On Rudimentarity, Primitive Recursivity and Representability
AU - Salehi, Saeed
TI - On Rudimentarity, Primitive Recursivity and Representability
AB - It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong)  representability of functions in sufficiently strong arithmetical theories.
VL - 2020
IS - Number 55
PY - 2020
SN - 0137-2904
C1 - 2084-2589
SP - 73
EP - 85
DO - 10.4467/20842589RM.20.004.12436
UR - https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/on-rudimentarity-primitive-recursivity-and-representability
KW - the incompleteness theorem
KW - bounded formulas
KW - rudimentary relations
KW - primitive recursive functions
KW - primitive recursive relations
KW - representability