User:IssaRice/Computability and logic/Some important distinctions and equivalences in introductory mathematical logic

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This page lists some important distinctions in introductory mathematical logic.

  • completeness: semantically complete (complete logic; the topic of the completeness theorem) vs negation-complete (complete theory; the topic of the first incompleteness theorem)
  • decides: deciding a sentence vs a theory being decidable vs deciding every sentence vs a logic being decidable
  • soundness: sound logic (soundness theorem) vs sound theory
  • truth in all interpretations (validity) vs truth in the intended interpretation (natural reading, standard interpretation)
  • structure vs interpretation vs model
  • theory vs axioms
  • \models: when a set of sentences comes before the symbol vs when a structure comes before the symbol