Lindström’s theorem

Lindstr\u00f6m's theorem is a significant result in metalogic, specifically concerning the properties of first-order logic. It demonstrates that first-order logic, characterized by properties like completeness and the L\u00f6wenheim-Skolem theorem, cannot be significantly extended while retaining these core features. The theorem essentially states that any logic that extends first-order logic and satisfies certain conditions (like the L\u00f6wenheim-Skolem theorem and either compactness or formal axiomatizability) will ultimately be equivalent to first-order logic itself. This implies that first-order logic is unique in its balance of expressive power and formal tractability.1