I have the following assignment question:
Find a first order sentence that is true in $\langle\mathbb{Q},+,\cdot ,0,1\rangle$ but not in $\langle \mathbb{R},+,\cdot,0,1\rangle$.
Most of what I can think of is second order (such as completeness, or cardinality).
I have the following candidate, though:
$\forall x\exists y\exists z \quad \Bigg(x=\bigg(\sum_{j=0}^y 1\bigg)\cdot i\bigg(\sum_{j=0}^z 1\bigg)\Bigg).$
Where $i(\alpha)$ denotes the multiplicative inverse of $\alpha.$
I'm just not sure that this idea of summing "$y$ times" or summing "$z$ times" is actually doable in first order logic. I'm also not sure if somehow I need to clarify that $y$ and $z$ need be integers (and if that's doable in FOL).
Any corrections or clarifications to my candidate are welcome, as well as maybe another example I'm just not thinking of.