ON THE DECIDABILITY OF THE THEORIES OF THE ARITHMETIC AND HYPERARITHMETIC DEGREES AS UPPERSEMILATTICES

Abstract

AbstractWe establish the decidability of the${{\rm{\Sigma }}_2}$theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with ≤, 0 , and$\sqcup$. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite uppersemilattices${\cal M}$and${\cal N}$, where${\cal M}$is a subuppersemilattice of${\cal N}$, that every embedding of${\cal M}$into either degree structure extends to one of${\cal N}$iff${\cal N}$is an end-extension of${\cal M}$.