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In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. [ 1] Second-order logic is in turn extended by higher-order logic and type theory . First-order logic quantifies only variables that range over individuals (elements of the domain of discourse ); second-order ...
By Gödel's completeness result, it must hence have a natural deduction proof (shown right). Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic . The completeness theorem applies to any first-order theory: If T is ...
For the earlier theory about the correspondence between truth and provability, see Gödel's completeness theorem. Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in ...
For example, every consistent theory in second-order logic has a model smaller than the first supercompact cardinal (assuming one exists). The minimum size at which a (downward) Löwenheim–Skolem–type theorem applies in a logic is known as the Löwenheim number, and can be used to characterize that logic's strength.
Proof theory is a major branch [ 1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Fagin's theorem. Fagin's theorem is the oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based descriptions of their problems rather than by the behavior of algorithms for solving those problems. The theorem states that the set of all properties ...
Stronger logics, such as second-order logic, are not complete. A consistency proof is a mathematical proof that a particular theory is consistent. [8] The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program.
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).