# The Kaufmann--Clote question on end extensions of models of arithmetic

## Mengzhou Sun

**Authors:** Mengzhou Sun, Tin Lok Wong and Yue Yang

A general question in the model theory of arithmetic is:

“for which theories \(S\), \(T\) and which \(n\in\mathbb N\) is it true that every countable sufficiently saturated model of a theory \(S\) has a proper \(\Pi_n\)-elementary end extension to a model of a theory \(T\)?”

Efforts over the past four decades have revealed answers
to this question for all the well-known theories in arithmetic,
i.e., the \(\Sigma_m\) induction schemes \(\mathrm I\Sigma_m\) and
the \(\Sigma_m\) collection schemes \(\mathrm B\Sigma_m\),
**except** the following instance.

Kaufmann [2] in the context of set theory; Clote [1]

“Is it true that, given any integer \(n\geqslant2\), every countable model of \(\mathrm B\Sigma_n\) has a proper \(\Pi_n\)-elementary end extension to a model of \(\mathrm B\Sigma_{n-1}\)?”

We present a positive answer to the Kaufmann–Clote question. The proof consists of a second-order ultrapower construction based on a low basis theorem. We will also include a survey on the answers related to the general question above.

## Bibliography

- P. Clote,
*Partition relations in arithmetic*,(C. A. Di Prisco, editor),Springer-Verlag,Berlin,1985,pp. 32–68.*Methods in Mathematical Logic* - M. Kaufmann,
*On existence of \(Sigma_n\) end extensions*,,(M. Lerman, J. H. Schmerl and R. I. Soare, editors),vol. 859,Springer,Berlin,1981,pp. 92–103.*Logic Year 1979–80*