The Borel complexity of the class of models of first-order theories

Key Points

• Models of first-order theories show specific Borel complexity, particularly under sharp conditions established for their classification.
• The analysis confirms that any sequential theory has a Pi omega 0 complete set of models, enhancing understanding of foundational structures.
• Using classical results, we provide criteria determining Borel completeness for various first-order theories in mathematical logic.
• This work emphasizes the critical nature of descriptive complexity in model theory, suggesting avenues for further research in foundational mathematics.