# Extended version of the theory of reals and its decidability

It is well-known due to Tarski that the theory of reals $(\mathbb{R},+,\cdot,<,=)$ is decidable. I was asking my self whether one would lose the decidability by adding all real constants. More formally, is the theory $(\mathbb{R},\{\underline{r} \mid r \in \mathbb{R}\} +,\cdot,<,=)$ evaluated in the standard model of the reals decidable? Here, the symbol $\underline{r}$ is such that its interpretation in the standard model is exactly the real $r$.

I guess that the extended version is still decidable, essentially because two reals are different if and only if there exists a rational number between them. Rationals, on the other hand, can be expressed in the usual Tarski fragment.

Can anyone help?

• How do you write a real number onto a tape which consists of $0$'s, $1$'s and blanks? – Andrej Bauer Jun 3 '15 at 21:01