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Many-sorted first-order logic allows variables to have different sorts, which have different domains. This is also called typed first-order logic, and the sorts called types (as in data type), but it is not the same as first-order type theory. Many-sorted first-order logic is often used in the study of second-order arithmetic. [33]
Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.
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.
For example, every consistent theory in second-order logic has a model smaller than the first supercompact cardinal (assuming one exists). The minimum size at which a (downward) Löwenheim–Skolem–type theorem applies in a logic is known as the Löwenheim number, and can be used to characterize that logic's strength.
There are three common ways of handling this in first-order logic: Use first-order logic with two types. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set". Use ordinary first-order logic, and instead of adding a new predicate to the language, treat "Set(t)" as an abbreviation for ...
A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z 2). Such subsystems are essential to reverse mathematics , a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength.
Second-order logic can be extended by a transitive closure operator in the same way as first-order logic, resulting in SO[TC]. The TC operator can now also take second-order variables as argument. SO[TC] characterises PSPACE. Since ordering can be referenced in second-order logic, this characterisation does not presuppose ordered structures. [20]
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables.