## Journal of Symbolic Logic

### Undecidable Extensions of Skolem Arithmetic

#### Abstract

Let $<_{P_2}$ be the restriction of usual order relation to integers which are primes or squares of primes, and let $\bot$ denote the coprimeness predicate. The elementary theory of $\langle\mathbb{N};\bot,<_{P_2}\rangle$, is undecidable. Now denote by $<_\Pi$ the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure $\langle\mathbb{N};\bot,<_\Pi\rangle$. Furthermore, the structures $\langle\mathbb{N};\mid,<_\Pi\rangle, \langle\mathbb{N};=,x,<_\Pi\rangle$ and $\langle\mathbb{N};=,+,x\rangle$ are interdefinable.

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 2 (1998), 379-401.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745507

Mathematical Reviews number (MathSciNet)
MR1625892

Zentralblatt MATH identifier
0911.03030

JSTOR
links.jstor.org

#### Citation

Bes, Alexis; Richard, Denis. Undecidable Extensions of Skolem Arithmetic. J. Symbolic Logic 63 (1998), no. 2, 379--401. https://projecteuclid.org/euclid.jsl/1183745507