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In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
The satisfiability problem for monadic second-order logic is undecidable in general because this logic subsumes first-order logic. The monadic second-order theory of the infinite complete binary tree, called S2S, is decidable. [8] As a consequence of this result, the following theories are decidable: The monadic second-order theory of trees.
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
Theory (mathematical logic) Complete theory. Vaught's test; Morley's categoricity theorem. Stability spectrum. Morley rank; Stable theory. Forking extension; Strongly minimal theory; Stable group. Tame group; o-minimal theory; Weakly o-minimal structure; C-minimal theory; Spectrum of a theory. Vaught conjecture; Model complete theory; List of ...
Recently, conservative extensions have been used for defining a notion of module for ontologies [citation needed]: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory. An extension which is not conservative may be called a proper extension.
This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test. More generally, any first-order theory with an infinite model has non-isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem.
The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory having an empty set of sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theories.
The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation . Examples