Fragments of existential second-order logic without 0-1 laws

We prove that there is a Monadic Σ₁¹ (Minimal Scott without equality) sentence without an asymptotic probability. Our result entails that the 0-1 law fails for the logics Σ₁¹(FO²) and Σ₁ ¹ (Minimal Godel without equality). Therefore we achieve the classification of first-order prefix classes with or without equality. According to the existence of the 0-1 law for the corresponding Σ ₁¹ fragment. In addition, our counterexample can be viewed as a single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known