Infinitary logic

Remarks

The logic $L_{\omega\omega}$ is first-order logic, since then all conjunctions, disjunctions and quantifier blocks are finite. The logic $L_{\omega_1\omega}$ allows countable conjunctions and disjunctions and finite quantifier blocks.

Example: in the signature of groups, the $L_{\omega_1\omega}$ sentence $\forall x \, \bigvee_{n \ge 1} (x^n = e)$ holds exactly in the torsion groups. Here $x^n$ is the term $x \cdot x \cdots x$ with $n$ factors.

Non-example: the expression $\forall x_1 \, \forall x_2 \, \forall x_3 \cdots \varphi$ with infinitely many quantified variables is not a formula of $L_{\omega_1\omega}$, because clause 4 allows only finite sets of variables when $\lambda = \omega$. It is a formula of $L_{\omega_1\omega_1}$ when $\varphi$ is.

The compactness theorem fails for $L_{\omega_1\omega}$, but the downward Löwenheim–Skolem theorem holds for its countable fragments.