A Simplified Proof of the 0–1 Law for Existential Second-Order Ackermann Sentences

Recently we gave a finitistic proof of the 0–1 law for ∑₁¹ (Ackermann) sentences, which relied as much as possible on the original argument of Kolaitis and Vardi. Here we present another version of our proof which, on the contrary, is self-contained. Finitism allows us to use the beautiful probabilistic argument of Kolaitis and Vardi in a simple and intuitive way. Consequently, we obtain a shorter proof.