axiomatic theory

Formal systems {axiomatic theory} can have primitives, definitions, axioms, and postulates in theory language {object language}.

primitive terms

Object language has undefined primitive symbols or objects.

definitions

Combining primitive terms defines further symbols or objects.

axioms

Combining primitives and definitions can make assumptions. Axioms assume identity element existence, inverse element existence, commutation law, association law, and distribution law. Axioms are independent of other axioms.

postulates

Object languages have valid statement structures and have logical rules for transforming statement structures. For example, variables can take values.

proofs

Starting from primitive terms, definitions, axioms, and postulates, logic can prove theorem or formula statements, by deduction and formal proofs.

examples

First-order predicate calculus {standard formalization} {first-order theory} is an example.

equivalences

The same axiomatic theory can use different symbols and relations {formulation} {model, axiomatic theory}. Axiomatic-theory primitive terms and relations can have different meanings {interpretation, axiomatic theory} {representation, axiomatic theory}.

Axiomatic theories using different symbols and relations can be formally the same {isomorphism, axiomatic theory}. Axiomatic theories can always have isomorphic forms {representation theorem}. Categorical theory shows how to prove that two axiomatic theories are isomorphic.

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Date Modified: 2022.0224