## Journal of Symbolic Logic

### Undecidable Extensions of Buchi Arithmetic and Cobham-Semenov Theorem

Alexis Bes

#### Abstract

Let $k$ and $l$ be two multiplicatively independent integers, and let $L \subseteq \mathbb{N}^n$ be a $l$-recognizable set which is not definable in $\langle\mathbb{N}; +\rangle$. We prove that the elementary theory of $\langle\mathbb{N}; +, V_k, L\rangle$, where $V_k(x)$ denotes the greatest power of $k$ dividing $x$, is undecidable. This result leads to a new proof of the Cobham-Semenov theorem.

#### Article information

Source
J. Symbolic Logic, Volume 62, Issue 4 (1997), 1280-1296.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745382

Mathematical Reviews number (MathSciNet)
MR1617949

Zentralblatt MATH identifier
0896.03011

JSTOR
links.jstor.org

#### Citation

Bes, Alexis. Undecidable Extensions of Buchi Arithmetic and Cobham-Semenov Theorem. J. Symbolic Logic 62 (1997), no. 4, 1280--1296. https://projecteuclid.org/euclid.jsl/1183745382