# Two Dichotomy Theorems

## TL;DR

This paper introduces the Dichotomy Property to analyze language expressivity in Set Computable Theory and proves undecidability results for certain language extensions involving MLS.

## Contribution

It defines the Dichotomy Property and uses it to establish undecidability of MLS extensions with specific operators.

## Key findings

- Dichotomy Property introduced for language analysis
- Undecidability proven for MLS with not ordered Cartesian product
- Undecidability proven for MLS with disjoint unary union operators

## Abstract

We introduce the Dichotomy Property, a new property of some languages in Set Computable Theory, in order to explore the expressivity of some languages which are extensions of MLS. By-product we prove undecidability of MLS extended with not ordered cartesian product and disjoint unary union operators.