compactness theorem

Target entity: compactness theorem
Context triple: [Löwenheim–Skolem theorem, relatedTo, compactness theorem]

• A. completeness theorem for first-order logic
  The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
• B. Hamilton’s compactness theorem
  Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
• C. Tarski–Mostowski–Robinson theorem
  The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
• D. Herbrand's theorem
  Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
• E. Löwenheim–Skolem theorem (via additional arguments)
  The Löwenheim–Skolem theorem is a fundamental result in model theory stating that any first-order theory with an infinite model has models of all infinite cardinalities, leading to the so-called Skolem paradox about the existence of countable models of set theory.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: compactness theorem
Target entity description: The compactness theorem is a fundamental result in mathematical logic stating that a set of first-order sentences has a model if and only if every finite subset of it has a model.

• A. completeness theorem for first-order logic
  The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
• B. Hamilton’s compactness theorem
  Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
• C. Tarski–Mostowski–Robinson theorem
  The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
• D. Herbrand's theorem
  Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
• E. Löwenheim–Skolem theorem (via additional arguments)
  The Löwenheim–Skolem theorem is a fundamental result in model theory stating that any first-order theory with an infinite model has models of all infinite cardinalities, leading to the so-called Skolem paradox about the existence of countable models of set theory.
• F. None of above. chosen