# What sets of real numbers are definable in the language of real closed fields?

The language of the first-order theory of real closed fields consists of the non-logical symbols $$0$$, $$1$$, $$+$$, $$\cdot$$, $$<$$, and $$=$$. My question is, for what subsets $$X$$ of $$\mathbb{R}$$ does there exist a formula $$\phi(x)$$ in the language of real closed fields such that $$X=\{x\in\mathbb{R}:\phi(x)\}$$?

I’ve heard people say that the semi-algebraic sets, i.e. finite unions of singletons and intervals, are what’s definable in the language of real closed fields. But I think they mean something slightly different than what I’m asking about. Because there are uncountably many singletons and intervals, but only countably many formulas.

• I mean, you can't define infinite fields on the reals at all, since they require infinite formulas. – Don Thousand Oct 18 '19 at 7:02
• What do you mean by define infinite fields? I’m just asking what subsets of the set of real numbers are first-order definable. – Keshav Srinivasan Oct 18 '19 at 7:02
• I suppose it was meant that the singletons and interval endpoint all need to be algebraic numbers – Hagen von Eitzen Oct 18 '19 at 7:12
• @HagenvonEitzen At least that’s not part of the definition of semialgebraic set: en.wikipedia.org/wiki/Semialgebraic_set But yeah, it does seem plausible that the answer to my question would be semialgebraic sets with algebraic endpoints. – Keshav Srinivasan Oct 18 '19 at 7:15
• @KeshavSrinivasan In other words, we talk about semialgebraic sets defined over the subring $\Bbb Z$ (or equivalently, over $\Bbb Q$). – Hagen von Eitzen Oct 18 '19 at 7:21

The definable subsets of $$\mathbb{R}$$ in the language $$\mathcal{L}_{\mathrm{or}}:=\{0, 1, +, \cdot, <\}$$ of ordered rings are unions of intervals (degenerated or otherwise) with algebraic endpoints, exactly as commented above by Hagen von Eitzen. A reference for this fact is Marker's Model Theory, An Introduction, Sect. 3.3.
There, it is proved that real closed fields have quantifier elimination in $$\mathcal{L}_{\mathrm{or}}$$, and the result follows from this.