TITLE: Why ZFC+FOL cannot uniquely describe/characterize R or N?
QUESTION [13 upvotes]: I find the following text on the Wikipedia page on first order logic:

First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

Here, what I want to ask is what does uniquely describe/characterize mean? Why is it that $\textbf{FOL}$ cannot uniquely describe/characterize $\mathbb{R}$ or $\mathbb{N}$?

REPLY [2 votes]: Still to some extent $\mathbb N$ is characterisable in FOL plus a small increment. The increment being the non-FOL requirement of being a least structure satisfying some list of axioms (Peano's axioms in the FOL version for $\mathbb N$). What about $\mathbb R$ ?