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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.

user105916
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  • All rational numbers are algebraic ($\frac ab$ is a root of $bx-a=0$), so it is hard to see how you could answer "yes" to (c) without also ansering "yes" to (b). – hmakholm left over Monica May 03 '16 at 20:47
  • I don't know anything about model theory, but for rational numbers, since you already have a definition for integers, maybe you can say any rational number $r$ is the solution to the equation $p=qr$ for some integers $p, q$. Also, for your definition of algebraic numbers, I'm pretty sure you have to restrict all of the $a_k$ to be rational. Otherwise, if the coefficients can be any real numbers, all real numbers would be algebraic by a polynomial in the form of $x-a$. – Noble Mushtak May 03 '16 at 20:48
  • Also, this previous discussion might help you. – Noble Mushtak May 03 '16 at 20:50
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    For (c), how do you differentiate between the different roots of a polynomial? So far all you have is ${x\in\mathbb R: f(x) = 0}$. – Mathmo123 May 03 '16 at 20:50
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    @NobleMushtak ${n}$ being definable is not the same as $\mathbb Z$ being definable. – Mathmo123 May 03 '16 at 20:52
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    I don't understand what $\varphi_2$ is trying to say – mercio May 03 '16 at 20:54
  • @HenningMakholm: But how can I tell, that $b$ and $a$ are rational numbers, when I have to define them in this task. – user105916 May 03 '16 at 20:55
  • @mercio: I wanted to seperate the three cases. $v_1$ can be positiv, negative or equal to 0. When $v_1$ is positiv, I can note it as sum $1+1+...+1$. If $v_1$ is negative, I write down $(-1)+...+(-1)$ and when $v_1=0$ there is nothing to do. – user105916 May 03 '16 at 20:56
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    @user105916: Um ... $a$ and $b$ in my comment are integers. By definition a rational number is one that equals $\frac ab$ for some integers $a$ and $b$. – hmakholm left over Monica May 03 '16 at 20:56
  • Oh, excuse me, I didnt get that right. But you are right, with this information it should be easy. Thanks. – user105916 May 03 '16 at 20:57
  • Regarding c, the $k$-largest root of a fixed polynomial (of fixed degree) is definable. The – Hagen von Eitzen May 03 '16 at 21:00
  • all the roots of a fixed polynomial of $\mathbb{Q}[x]$ are definable, for example you can say that "it is the root which is the closest to $\frac{a}{b} +i \frac{c}{d}$ in $|.|$" – reuns May 03 '16 at 21:30

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