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List of first-order theories
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties.
For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure.
There are two common ways to specify theories:
An Lσ theory may:
The signature of the pure identity theory is empty, with no functions, constants, or relations.
Pure identity theory has no (non-logical) axioms. It is decidable.
One of the few interesting properties that can be stated in the language of pure identity theory is that of being infinite. This is given by an infinite set of axioms stating there are at least 2 elements, there are at least 3 elements, and so on:
These axioms define the theory of an infinite set.
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List of first-order theories
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties.
For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure.
There are two common ways to specify theories:
An Lσ theory may:
The signature of the pure identity theory is empty, with no functions, constants, or relations.
Pure identity theory has no (non-logical) axioms. It is decidable.
One of the few interesting properties that can be stated in the language of pure identity theory is that of being infinite. This is given by an infinite set of axioms stating there are at least 2 elements, there are at least 3 elements, and so on:
These axioms define the theory of an infinite set.