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Define $x\in\mathbb{R}$ to be arithmetical number if the set $\{\langle p, q \rangle \in \mathbb{Z}^2 : \frac{p}{q} < x\}$ is an arithmetical set. Define $x\in\mathbb{C}$ to be arithmetical number if its real and imaginary parts are both arithmetical real numbers. This is very wide (although still countable) class of numbers: for example, every computable number is arithmetical, and non-computable Chaitin's constant Ω is also arithmetical.

It is quite obvious that sum and difference of two arithmetical numbers is also arithmetical. What we can say about their products, exponents, roots of polynomials with arithmetical coefficients, volumes of regions in $\mathbb{R}^n$ defined by systems of polynomial inequalities with arithmetical coefficients?

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The recent "burst" of questions here and on MO makes me wonder what is your ultimate goal... :-) – Asaf Karagila Dec 7 '11 at 22:59
up vote 5 down vote accepted

Yes, all those quantities remain arithmetical.

The class of arithmetic reals is really quite robust, and it can be described in a variety of equivalent ways. For example, any subset of $\mathbb{N}$ that can be described in the first order structure $\langle\mathbb{N},+,\cdot,0,1,\lt\rangle$ is arithmetic, and we can go back and forth between subsets of $\mathbb{N}$ and individual reals by means of their binary expansions, whose basic properties can be treated in this language.

All the quantities in your examples can be described by quantifying only over the natural numbers in the manner required. For example, one can characterize each of your numbers as a limit of certain rational approximations, which can be easily described without any need to quantify over all reals.

A more general setting for arithmetic sets applies not only to individual reals or sets of natural numbers, but also to sets of reals: A subset $A\subset\mathbb{R}$ is arithmetic if it can be defined in the structure of the reals, where the language includes a predicate for the integers and basic arithmetic operations that allow one access to the digits of the reals, but by a formula using quantification only over the natural numbers, but not over the reals. Allowing the definitions to quantify over the reals brings one into the projective hieararchy, which sits atop the arithmetic hierarchy.

The kind of thing that brings you out of the class of arithmetic sets is when you need to quantify not only over the natural numbers, but over the reals. For example, the projection of an arithmetic subset of the real plane to one of the coordinate axes is a process that takes one up to the level of analytic sets, or $\Sigma^1_1$. Similarly, the continuous image of an arithmetic set of reals need not be arithmetic.

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