Definability of transcendental reals in the real field by a single formula

Question

This is a follow up to my previous question on definability of algebraic numbers in the real field. In that question, I observed that in the structure $(\mathbb{R},+,-,*,0,1,\leq)$, the set of transcendental real numbers is definable by an infinite set of formulas. Specifically, we just have an axiom, for each polynomial equation with rational coefficients, that the number does not satisfy it. I am wondering if it is possible to use a finite set of formulas, or equivalently, a single formula, to define the set of transcendentals.

Answer 1

As already linked in the comments: technically this answer is enough. The property that is described there is o-minimality. That means that every definable set (in one variable) in a real closed field (such as $(\mathbb{R}; +, -, *, 0, 1, <)$) is a finite union of open intervals and points. Clearly this means that the transcendental numbers cannot be definable (not even with parameters).

As in the linked answer, a reference is Marker's Model Theory: An Introduction section 3.3. In particular Theorem 3.3.15 proves quantifier elimination from which o-minimality (Corollary 3.3.23) easily follows.