Why do metalogicians study formal systems?

Metalogicians study formal systems to analyze the deductive structure of thought and to investigate the fundamental properties of logical and mathematical frameworks. By abstracting away from specific meanings, these systems provide an ideal language for examining the relationship between syntax (the arrangement of symbols) and semantics (the interpretation of those symbols).12

Key reasons for this study include:

• Analyzing Deduction: Formal systems serve as tools to understand the concept of deduction, allowing researchers to define axioms, rules of inference, and theorems with precision.1
• Evaluating Systemic Properties: Metalogicians focus on the completeness and consistency of these systems. This research led to Kurt Gödel’s discovery that in significant formal systems, not all true sentences are provable as theorems.34
• Defining Computability: The study of formal systems has helped clarify concepts such as "mechanical" and "computable," leading to the development of recursion theory and the identification of problems that are mechanically unsolvable.4
• Foundational Inquiry: These systems form the basis for exploring the foundations of mathematics and provide a framework for analyzing deductive sciences.1