# Computable quotient presentations of models of arithmetic and set theory

## TL;DR

This paper extends the Tennenbaum phenomenon to computable quotient presentations, showing that no nonstandard models of arithmetic or set theory can have such presentations with computable or c.e. equivalence relations.

## Contribution

It proves that various classes of nonstandard models of arithmetic and set theory cannot have computable quotient presentations, generalizing the Tennenbaum phenomenon to broader contexts.

## Key findings

- No nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation.
- No nonstandard model of arithmetic with $\Sigma_1$-soundness has a computable quotient presentation by a co-c.e. equivalence relation.
- No model of ZFC or weaker set theories has a computable quotient presentation by any equivalence relation.

## Abstract

We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No $\Sigma_1$-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language $\{+,\cdot,\leq\}$ has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no nonstandard model of finite set theory has a computable quotient presentation.