Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

Abstract

We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably recursive functions of BA is a proper sub-class of primitive recursive functions. Three extensions of BA, called BA + U, BAc and EBA are investigated with relation to their provably recursive functions. It is shown that the provably recursive functions of these three extensions of BA are exactly primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem doesn't hold in BA, BA + U, BAc, but holds in EBA.

Not all Kripke models of $\sf HA$ are locally $\sf PA$

“…One of these logics is Visser's Basic logic, and its extension Extended Basic logic. The model theory of arithmetic over these logics were investigated in [13,5,6]. From the point of view of Problem 1.1, it is proved in [4] that every irreflexive node in a Kripke model of BA (Basic Arithmetic) is locally I∃ + 1 .…”

“…From the point of view of Problem 1.1, it is proved in [4] that every irreflexive node in a Kripke model of BA (Basic Arithmetic) is locally I∃ + 1 . So In general, every irreflexive node in a Kripke model of the natural extension of BA such as EBA (Extended Basic Arithmetic) is locally IΣ 1 (see Corollary 3.33 in [6]). Also it is proved in [6] that every Kripke model of EBA is locally Th Π2 (IΣ 1 ) + Th Π1 (PA).…”

“…So In general, every irreflexive node in a Kripke model of the natural extension of BA such as EBA (Extended Basic Arithmetic) is locally IΣ 1 (see Corollary 3.33 in [6]). Also it is proved in [6] that every Kripke model of EBA is locally Th Π2 (IΣ 1 ) + Th Π1 (PA). Note that every Kripke model of HA is also a Kripke model of BA and…”