We investigate the infinitary logic L∞ωω, in which sentences may have arbitrary disjunctions and conjunctions, but they involve only finite numbers of distinct variables. We show that various fixpoint logics can be viewed as fragments of L∞ωω, and we describe a game-theoretic characterization of the expressive power of the logic. Finally, we study asymptotic probabilities of properties expressible in L∞ωω on finite structures. We show that the 0-1 law holds for L∞ωω, i.e., the asymptotic probability of every sentence in this logic exists and is equal to either 0 or 1. This result subsumes earlier work on asymptotic probabilities for various fixpoint logics and reveals the boundary of 0-1 laws for infinitary logics. © 1992.