If $\mathcal{L}$ is a countable, first-order language, it is easy to see that every $n$-type $p$ (over $\emptyset$) can be expressed as an $\mathcal{L}\_{\omega\_1,\omega}$-formula, namely $\bigwedge\_{\psi\in p} \psi$. Does the converse hold? Namely, given an $\mathcal{L}\_{\omega\_1,\omega}$ formula $\psi(v\_1,\ldots,v\_n)$ is there an $n$-type $p$ such that $\mathfrak{A}\models\psi(\bar{a})$ if and only if $\bar{a}$ realizes $p$ in $\mathfrak{A}$?


On a related note, are there any nice canonical forms for such infinitary formulas?



Can every $\mathcal{L}_{\omega_1,\omega}$ formula be expressed as a type? What about canonical forms?
No. In the language of first-order arithmetic, with $[n]$ denoting the standard numeral for $n$, the formula $\bigvee\_{n\in\mathbb N}(v=[n])$ expresses that $v$ is a standard natural number. There is no type (conjunction of first-order formulas) that is equivalent to it in all models of first-order arithmetic. Proof: If there were such a type, then the theorems of first-order arithmetic, the formulas in the alleged type, and the formulas $\neg(v=[n])$ would form an inconsistent set of first-order formulas with no finite inconsistent subset, contrary to the compactness theorem of first-order logic.


