Random Graph: Stronger logic but with the zero one law

TL;DR

This paper introduces a new logic stronger than first-order logic that still satisfies the zero-one law for the random graph with edge probability 1/2, revealing new logical properties of random graphs.

Contribution

It demonstrates the existence of a logic stronger than first-order logic that maintains the zero-one law in the context of random graphs.

Findings

Identifies a logic stronger than first-order logic with the zero-one law.

Shows there exists a formula not equivalent to any first-order formula in the random graph.

Establishes that stronger logics can satisfy the zero-one law in random graph models.

Abstract

We find a logic really stronger than first order for the random graph with edge probability $\frac 12$ but satisfies the 0-1 law. This means that on the one hand it satisfies the 0-1 law, e.g. for the random graph ${\mathcal G}_{n,1/2}$ and on the other hand there is a formula $\varphi(x)$ such that for no first order $\psi(x)$ do we have: for every random enough ${\mathcal G}_{n,1/2}$ the formulas $\varphi(x),\psi(x)$ equivalent in it.