Decidability of A Theory
A theory is a set of formulas, which here is assumed to be closed under logical consequence. The question of decidability for a theory is whether there is an effective procedure that, given an arbitrary formula in the signature of the theory, decides whether the formula is a member of the theory or not. This problem arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms. Examples of decidable first-order theories include the theory of real closed fields, and Presburger arithmetic, while the theory of groups and Robinson arithmetic are examples of undecidable theories.
There are several basic results about decidability of theories. Every inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus member of, the theory. Every complete recursively enumerable first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.
A consistent theory which has the property that every consistent extension is undecidable is said to be essentially undecidable. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable. Robinson arithmetic is known to be essentially undecidable, and thus every consistent theory which includes or interprets Robinson arithmetic is also (essentially) undecidable.
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Famous quotes containing the word theory:
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—Ralph Waldo Emerson (1803–1882)