  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.
