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

We prove that there is a Monadic Sigma(1)(1) (Minimal Scott without equality) sentence without an asymptotic probability. Our result entails that the 0-1 law fails for the logics Sigma(1)(1) (FO2) and Sigma(1)(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 Sigma(1)(1) 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.