Computable quotient presentations of models of arithmetic and set theory
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 -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 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.
Keywords
Cite
@article{arxiv.1702.08350,
title = {Computable quotient presentations of models of arithmetic and set theory},
author = {Michał Tomasz Godziszewski and Joel David Hamkins},
journal= {arXiv preprint arXiv:1702.08350},
year = {2017}
}
Comments
This is a preliminary report of research initiated at the conference Mathematical Logic and its Applications, held September 2016 in memory of Professor Yuzuru Kakuda of Kobe University at the Research Institute for Mathematical Sciences (RIMS) in Kyoto. 15 pages. Commentary can be made at http://jdh.hamkins.org/computable-quotient-presentations