Abstract

We show that the 0-1 law does not hold for the class $\Sigma^1_1 $ by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class $\Sigma^1_1 \forall\exists\forall$ with equality.