1. A property $\Phi$ that corresponds to a sentence $\phi$ in a formal language that can be captured in a first-order language, that is

• • $\mathfrak{M}$ satisfies $\Phi$ if and only if $\phi$ is true in $\mathfrak{M}$.

Some of the simplest first-order properties in classical logic are those corresponding to models with $n$ elements for a natural number $n$. For example, the property that holds of models with precisely three elements can be captured by the sentence of first-order logic with identity:

• • $\exists x\exists y\exists z\left(\neg \left(x=y\right)\wedge \neg \left(x=z\right)\wedge \neg \left(y=z\right)\wedge \forall u\left(u=x\vee u=y\vee u=z\right)\right)$

Examples of properties not first-order in classical logic include ‘$\mathfrak{M}$ has finitely many elements’ and ‘$\mathfrak{M}$ has $\kappa$ many elements’ for an infinite cardinal $\kappa$. In first-order classical logic, there exists no sentence $\phi$ that is made true in all and only models with finite domains and by the upward Löwenheim-Skolem theorem, if $\phi$ has a model of size $\kappa$, then it has models of size $\lambda$ for all $\lambda >\kappa$.

2. A property $\Phi$ for which there exists a formula $\psi \left(x\right)$ such that in any model $\mathfrak{M}$ and elements $a$ in its domain,

• • $\Phi$ is true of $a$ in $\mathfrak{M}$ if and only if $a$ satisfies $\Phi \left(x\right)$ in $\mathfrak{M}$.

Properties that are first-order in this sense are sometimes referred to as firstorderizable.