. Any system wherein certain expressions are derived in accordance with a given set of rules from a decidable initial set of expressions taken as given (and called axioms). The axioms themselves of such a system form an axiom set. ‘Axiom system’ is often used for ‘axiom set’. The formation rules specify what elements or symbols the system is going to use and what combinations of them are to count as expressions which can serve as axioms or be tested to see whether they can be derived from the axioms. These expressions are called well-formed formulae or wff, for short, and those of them which can be derived from the axioms are called theorems. The formation rules are analogous to rules of grammar, and the wff analogous to meaningful sentences. The axioms themselves will count as theorems if, as in most systems, they are trivially derivable from themselves. For reasons of economy and elegance the axioms should be independent, i.e. not derivable within the given system from each other. The axioms may be infinite in number, provided rules for selecting them are given.
Such a rule will define an axiom scheme by saying ‘All wff of such and such a kind are to count as axioms’. The transformation rules say what wff can be derived from others, and so govern what the theorems of the system will be, given the axioms.
In an abstract axiom system the expressions are simply symbols, or marks on paper. But if the system is applied to a certain subject-matter we have a MODEL or interpretation of the system, and the subject-matter is said to be axiomatized. To axiomatize a subject is thus to systematize it, and show how most of it can be derived if certain selected axioms and transformation rules are taken for granted. These are so selected that the system shall be CONSISTENT and, where possible, COMPLETE. The axioms are therefore either true propositions, which need not be simple or obvious, or propositions which can be postulated as true without leading to contradiction, as in non-Euclidean geometries (see SPACE). The transformation rules are related to VALIDITY as the axioms are to truth. See also MODELS, BOOLEAN ALGEBRA.
C.Glymour, Thinking Things Through, MIT Press, 1992.
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