# 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 and computability theory. Bizarrely, most books won't even mention these distinctions, so you will probably be very confused at the start as you inevitably conflate these ideas.

• 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 (decidability of logic has several equivalent formulations; this is the topic of the Entscheidungsproblem)
• 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
• language vs logic vs formal system vs system
• proves logically vs proves in a particular theory
• $\models$: when a set of sentences comes before the symbol vs when a structure comes before the symbol
• deciding (corresponding to computable total functions) vs recognizing (semi-deciding, computably enumerating; corresponding to computable partial functions)
• algorithm vs program vs index vs Godel number
• expresses vs captures (strong capture, weak capture)
• primitive recursive vs (general, μ) recursive
• $\Delta_0 = \Sigma_0 = \Pi_0$ vs $\Delta_1 = \Sigma_1 \cap \Pi_1$
• $\Sigma_n$ formulas vs relations vs sets

More speculative:

• different kinds of diagonalization?