Consider the structure $(\mathbb{R},+,-,*,0,1,\leq)$. We adjoin to it a constant $r$. Is there a set $S$ of formulas in that expanded language, perhaps an infinite set, such that the members of $S$ are jointly satisfied iff $r$ is an algebraic real number? Certainly, an infinite set is possible for defining the transcendental numbers.
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No such set $S$ can exist. The (real) algebraic numbers $\mathbb{R}_\text{alg}$ is an elementary substructure of $\mathbb{R}$. So any formula $\phi(x)$ that is satisfied by all algebraic numbers must be satisfied by all elements in the structure.
You can find the above claim in Marker's Model Theory: An Introduction, Corollary 3.3.16.
Mark Kamsma
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