I have to solve the following task and got some problems with it:
a) Be $n\in\mathbb{Z}$. Is $\{n\}$ definable over $(\mathbb{R}, <, +, \cdot, 0, 1)$
b) Be $q\in\mathbb{Q}$. Is $\{q\}$ definable over $(\mathbb{R}, <, +, \cdot, 0, 1)$?
c) Be $x\in\mathbb{R}$ algebraic. Is $\{x\}$ definable over $(\mathbb{R}, <, +, \cdot, 0, 1)$?
I think that a) and c) should be definable, but I am not so sure about b) yet. Also I do not know exactly, if the way I am going to give the formula, is correct.
a)
I think $\{n\}$ is definable, because I can get every integer by adding $1$ or $(-1)$ only, excluding the 0.
So I could first give a formula, which describes the $(-1)$ and then use it.
$\varphi_1\equiv ((v_0\cdot v_0=1)\wedge (v_0\neq 1))$
$\varphi_2\equiv (v_1<0(v_1=v_0+\dotso+v_0)\vee (0<v_1(v_1=1+\dotso+1))\vee (v_1=0))$
I could combine $\varphi_1$ and $\varphi_2$ if needed.
To c)
Since $x=v_0$ is supposed to be an algebraic number, it is the solution of a polynomial equation
$a_kv_0^k+a_{k-1}v_0^{k-1}+\dotso+a_1v_0+a_0=0$
Of course I can note $v_0^l$ as $v_0\cdot v_0\cdot\dotso\cdot v_0$, but it is the notation with the dots, I really do not like, and I dont know if I am allowed to use it.
Are my thoughts correct? Is b) definable as well. I am inclinde to think so, but I am not quite sure yet.
I would be interested in your thoughts about this. Thanks in advance.