# Truncation in Hahn Fields is Undecidable and Wild

## TL;DR

This paper proves that truncation in Hahn fields leads to undecidability and complex logical properties, revealing the wild logical behavior of such structures.

## Contribution

It demonstrates that Hahn fields with truncation can interpret monadic second-order logic, making their theory undecidable and exhibiting SOP and TP properties.

## Key findings

- Hahn fields with truncation are undecidable.
- A definable relation with SOP and TP is constructed.
- Interpretation of monadic second-order logic in Hahn fields.

## Abstract

We show that in any nontrivial Hahn field with truncation as a primitive operation we can interpret the monadic second-order logic of the additive monoid of natural numbers and are thus undecidable. We also specify a definable binary relation on such a structure that has $\SOP$ and $\TP$.