I have seen in lots of texts that the theory of $(\Bbb{N},<)$ does not admit quantifier elimination. I am trying to prove why, and although I can find hints in the literature, I am struggling to complete the proof.
I think the best method to do this would be to look at sentences in one variable $x$, and show that all of these sentences either admit the value True for all $x \in \Bbb{N}$ or False for all $x \in \Bbb{N}$. I can see that all atomic formulas made up of one variable will be of the form $x=x$ or $x<x$, and that any Boolean combinations of these atomic formulae will take the value or true or false for the entire domain. I think it therefore remains for me to find a sentence $\phi(x)$, that contains quantifiers, and only contains the one variable $x$ but is satisfied by a unique element of the domain, or a unique subset of the domain, such that it cannot be expressed by a Boolean combination of atomic formulas in one variable. My question is therefore, what should () be? Any help or hints appreciated, thank you!