An ambitious project by David Hilbert to formalize all mathematics and prove its consistency using finitary methods. It aimed to…
A higher-order variable represents functions, predicates, or relations, distinguishing it from variables that denote individual objects. This concept is fundamental…
A Henkin sentence is a self-referential statement that asserts its own provability within a formal system. It's a foundational concept…
Gödel's second incompleteness theorem states that no consistent formal system strong enough to include basic arithmetic can prove its own…
Gödel's First Incompleteness Theorem states that any consistent formal system capable of basic arithmetic contains true statements that are unprovable…
A self-referential sentence in formal systems, a Gödel sentence demonstrates incompleteness theorems by asserting its own unprovability within that system.…
Gödel numbering assigns unique natural numbers to symbols, formulas, and proofs in formal systems. This allows mathematical statements to be…
A first-order theory formalizes mathematical reasoning using first-order logic. It defines relationships between individuals, properties, and relations, forming the foundation…
A theory is finitely axiomatizable if it can be completely defined by a finite collection of fundamental statements or axioms.…
Finite character describes systems where all essential properties can be understood by analyzing a limited, finite portion. This concept is…