Counterexamples of the 0-1 Law for Fragments of Existential Second-Order Logic: An Overview

Bulletin of Symbolic Logic 6 (1):67 - 82 (2000)
Abstract
We propose an original use of techniques from random graph theory to find a Monadic ∑ 1 1 (Minimal Scott without equality) sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics ∑ 1 1 (FO 2 ) and ∑ 1 1 (Minimal Gödel without equality). Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the corresponding ∑ 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.
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