# Explicit sentences distinguishing McDuff's II$_1$ factors

## TL;DR

This paper provides explicit sentences that distinguish McDuff's II$_1$ factors, building on prior work that proved their non-isomorphism and non-elementary equivalence using ultrapowers and Ehrenfeucht-Fra"isse games.

## Contribution

It introduces concrete sentences that explicitly distinguish McDuff's II$_1$ factors, advancing beyond previous bounds on quantifier complexity.

## Key findings

- Explicit sentences distinguishing McDuff's factors are constructed.
- The results confirm non-isomorphism through concrete logical sentences.
- Addresses open question about concrete distinguishing sentences.

## Abstract

Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\"isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.