Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA is $\omega$-inconsistent and all infinite models are non-standard.

Is any ACF a recursive model of MA? If so, how would such a model avoid Tennenbaum's theorem? I know there are weak theories of arithmetic with recursive non-standard models, but MA has the same induction schema as PA.

I have read "An old theorem of A. J. Wilkie (Some results and problems on weak systems of arithmetic, Logic Colloquium '77) asserts that a discretely ordered ring $R$ can be extended to a model of open induction if and only if for all $n>1$, there is a homomorphism from $R$ onto $\mathbb{Z} /n \mathbb{Z}$". An ACF is not a discretely ordered ring, but any $\mathbb{Z} /n \mathbb{Z}$ is a model of MA. Is $\mathbb{A}$ homomorphic to some $\mathbb{Z} /n \mathbb{Z}$ or are ACF's different from other models of MA?

I asked this question on SE and got no answer.

The standard models of MA are the rings $\mathbb{Z} /n \mathbb{Z}$ where $\mathbb{Z}$ are the standard integers and $n$ is a standard natural number. All standard models of MA are finite. Let $\mathbb{N}^*$ be a countable non-standard model of PA and let $\mathbb{Z}^*$ be the integers extended from $\mathbb{N}^*$. A "non-standard" model of MA would be a ring $\mathbb{Z}^* /n^* \mathbb{Z}^*$ where $n^*$ is a non-standard natural number larger than any standard natural number.

We know the structure of $\mathbb{N}^*$ is $\omega + (\omega ^* + \omega) \cdot \eta$ where $\omega$ is the order type of the natural numbers, $(\omega ^* + \omega)$ is the order type of the integers and $\eta$ is the order type of the rationals. The $(\omega ^* + \omega)$ structures are sometimes call Z-blocks. The usual argument for the structure of countable non-standard models starts by assuming $a$ and $b$ are natural numbers and $|b-a|$ is larger than any standard natural number. Assume $a+b$ is even. Then $\frac{a+b}{2}$ must be an infinite distance from $a$ and an infinite distance from $b$. This shows the Z-blocks are dense. Between any two Z-blocks there is another Z-block.

I have always assumed similar arguments show a ring $\mathbb{Z}^* /n^* \mathbb{Z}^*$ has the structure $(\omega ^* + \omega) \cdot \eta$. I expand my question to ask if there are recursive models of $\mathbb{Z}^* /n^* \mathbb{Z}^*$.

Everycountable ACF is isomorphic to a recursive one. Tennenbaum’s theorem is about models of PA (or its weak subsystems like $IE_1$), so it does not apply to MA. I don’t understand the last question. What is $\mathbb A$? If it is supposed to be an ACF, then no, because every homomorphism from a field into another ring is automatically an embedding. $\endgroup$ – Emil Jeřábek Dec 14 '13 at 18:15shouldexpect recursive sets to be encodable in the same way as in PA in a theory which is quite different from PA. Pretty muchnothingin usual proofs of Tennebaum theorem works for MA. What you call for whatever reason “nonstandard models of MA” do not behave in any useful way similarly to nonstandard models of PA. The main property of nonstandard models of PA is that they are end-extensions of the standard model under the ordering of the model, and then induction gives overspill. None of this makes any sense without an ordering. $\endgroup$ – Emil Jeřábek Dec 14 '13 at 18:33