Infinitary logic

Summary

An infinitary logic is an extension of first-order logic that allows formulas of infinite length. Examples are conjunctions of infinitely many formulas and infinite blocks of quantifiers. The logics $L_{\kappa\lambda}$ are the standard family, and $L_{\omega_1\omega}$, with countable conjunctions and finite quantifier blocks, is the most studied one. These logics can express properties that first-order logic cannot, but they lose the compactness theorem.