6

In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying : $$\text{for all } x \in \Bbb R, \quad x \in \Bbb R_+ \;\text{ if and only if} \;\; \phi[x] $$

My question is : what happens if I replace $\Bbb R$ by $\Bbb Q$ ? More precisely :

Is there a first-order formula $\phi[x]$ of $\scr L$, such that $$\text{for all } x \in \Bbb Q, \quad x \in \Bbb Q_+ \;\text{ if and only if} \;\; \phi[x] $$

Said differently, I would like to know if the set of the positive rationals is definable in that language. Related questions are, for instance : (1), (2).

I don't know if the $\scr L$-structure $(\Bbb Q, +, \cdot, 0)$ admits elimination of quantifiers. If this is the case, then this could be helpful ; see this answer.

Thank you for your comments !

Community
  • 1
Watson
  • 23,793

2 Answers2

13

Yes. Every rational number $\ge 0$ is the sum of four squares. This is easily derived from Lagrange's Theorem which says that every non-negative integer is the sum of four squares.

Note that the formula we get is existential.

André Nicolas
  • 507,029
  • Thank you for your nice answer ! That may be very easy, but I don't immediately see how you can derive this from Lagrange's Theorem. – Watson Feb 03 '16 at 21:06
  • 3
    Let $r$ be a non-negative rational. Then $r=\frac{a}{b}$ where $a$ and $b$ are integers, with $b\gt 0 $. So $r=\frac{ab}{b^2}$. Now express $ab$ as a sum of four squares $s^2+t^2+u^2+v^2$. Then $r=(s/b)^2+(t/b)^2+(u/b)^2+(v/b)^2$. – André Nicolas Feb 03 '16 at 21:10
2

It is a famous result of Julia Robinson that $(\Bbb{Q}, +, \cdot, 0)$ is undecidable. This implies that $(\Bbb{Q}, +, \cdot, 0)$ does not admit elimination of quantifiers. That the rational numbers are not definable in the first-order theory of the reals follows from this, but also follows from well-known facts about O-minimality of the first-order theory of the reals

Rob Arthan
  • 48,577
  • Thank you for your answer. You gave an answer to my "I don't know if the structure $(\Bbb Q,+,\cdot,0)$ admits elimination of quantifiers", but how is the part of your sentence "the rational numbers are not definable in the first-order theory of the reals" linked to my question ? I apologize if this is trivial, but I don't see how these are related. Anyway, I've learned something :-) – Watson Feb 04 '16 at 15:47
  • 1
    It was just an extra bit of information that I thought you mind find interesting in this context. – Rob Arthan Feb 04 '16 at 20:45