Archimedean ordered field and non-expressible in first order logic In Chang and Keisler's model theory, they define that an ordered field $(F,+_{F},\cdot,0_{F},1_{F},\le)$ is Archimedean iff for any two positive elements $a$, $b$ in $F$ there is an $n$ such that $b\le na$.
(Here the notation $na$ is an abbreviation for the term $a+_{F}\ldots+_{F}a$, $n$ times.)
And they say that this definition is not expressible in first order logic.
My question is why it is impossible to express the definition in first order logic. In $\mathrm{ZF}$, we can define recursively a function $f$ such that $\mathrm{dom}(f)=\omega$, $f(0)=a$ and $f(n+1)=f(n)+_{F}a$. So the definition can be formalized in $\mathrm{ZF}$. I don't know what I'm missing..
 A: As Alex Kruckman says, the intended meaning is that this definition can't be expressed in the first-order language of ordered fields. The problem is that the first-order language of ordered fields does not obviously allow you to quantify over all positive integers $n$; you can talk about
$$na = \underbrace{a + \dots + a}_{n \text{ times}}$$
for any fixed $n$, but the first-order language of ordered fields $F$ only allows you to quantify over $F$, and doesn't obviously supply you with a way of talking about $\mathbb{N} \subset F$.
I say "obviously" because a priori there could be a clever trick that does it, but in fact there can't be such a trick. This is because the class of models of a first-order theory must be closed under ultraproducts, but it's possible for an ultraproduct of archimedean fields to be non-archimedean. For example the ultrapower $\mathbb{R}^{\infty}/U$ of countably many copies of $\mathbb{R}$ (the nonstandard reals) is non-archimedean: we can take $a = (1, 1, 1, \dots)$ and $b = (1, 2, 3, \dots)$ and then there is no positive integer $n$ such that $b \le na$ (in other words, $b$ is infinitely large compared to $a$).
Ignorable additional comment: You can extend the first-order theory of ordered fields to a two-sorted theory consisting of one sort for the natural numbers $\mathbb{N}$ and another sort for the field $F$, together with, say, an embedding $\mathbb{N} \hookrightarrow F$; this would allow you to talk about the operation $(n, a) \mapsto na$ where $n \in \mathbb{N}$ and $a \in F$, and to quantify over $\mathbb{N}$. Now the problem is that it's impossible to write down axioms guaranteeing that the first sort is exactly $\mathbb{N}$ and not some larger nonstandard model of the natural numbers; for example, whatever axioms we write down, we can take an ultrapower of the pair $(\mathbb{N}, \mathbb{R})$ and we'll have infinitely large nonstandard natural numbers, so the ultrapower $\mathbb{R}^{\infty}/U$ will still be "archimedean" but with respect to the nonstandard natural numbers $\mathbb{N}^{\infty}/U$, not the standard ones. So this doesn't work either.
