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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 ...
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original.
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.
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 ...
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member.
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
Let A be a non-empty set, X a subset of A, F a set of functions in A, and + the inductive closure of X under F. Let be B any non-empty set and let G be the set of functions on B, such that there is a function : in G that maps with each function f of arity n in F the following function (): in G (G cannot be a bijection).
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). [ 1 ] Existence