- Set theories
- The usual signature of set theory has one binary relation ∈, no constants, and no functions. Some of the theories below are "class theories" which have two sorts of object, sets and classes. There ar ...

- Second order arithmetic
- Second-order arithmetic can refer to a first order theory (in spite of the name) with two types of variables, thought of as varying over integers and subsets of the integers. (There is also a theory o ...

- Arithmetic
- Many of the first order theories described above can be extended to complete recursively enumerable consistent theories. This is no longer true for most of the following theories; they can usually enc ...

- Addition
- The theory of the natural numbers with a successor function has signature consisting of a constant 0 and a unary function S ("successor": S(x) is interpreted as x+1), and has axioms:∀x ¬ Sx = 0∀x∀ ...

- Differential algebra
- The theory DF of differential fields.The signature is that of fields (0, 1, +, -, ×) together with a unary function ∂, the derivation. The axioms are those for fields together with\forall u\forall v ...

- Geometry
- Axioms for various systems of geometry usually use a typed language, with the different types corresponding to different geometric objects such as points, lines, circles, planes, and so on. The signat ...

- Rings and fields
- The signature of (unital) rings has 2 constants 0 and 1, two binary functions + and ×, and, optionally, one unary inverse functions − −1.Rings Axioms: Addition makes the ring into an abelian group, ...

- Groups
- The signature of group theory has one constant 1 (the identity), one function of arity 1 (the inverse) whose value on t is denoted by t−1, and one function of arity 2, which is usually omitted from t ...

- Boolean algebras
- There are several different signatures and conventions used for Boolean algebras:The signature has 2 constants, 0 and 1, and two binary functions ∧ and ∨ ("and" and "or"), and one unary function ¬ ...

- Graphs
- The signature of graphs has no constants or functions, and one binary relation symbol R, where R(x,y) is read as "there is an edge from x to y".The axioms for the theory of graphs areSymmetric: ∀x ∀ ...

- Lattices
- Lattices can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature consisting of tw ...

- Orders
- The signature of orders has no constants or functions, and one binary relation symbols ≤. (It is of course possible to use ≥, or instead as the basic relation, with the obvious minor changes to th ...

- Equivalence relations
- The signature of equivalence relations has one binary infix relation symbol ~, no constants, and no functions. Equivalence relations satisfy the axioms:Reflexivity ∀x x~x;Symmetry ∀x ∀y x~y → y~x; ...

- Unary relations
- A set of unary relations Pi for i in some set I is called independent if for every two disjoint finite subsets A and B of I there is some element x such that Pi(x) is true for i in A and false for i i ...

- Pure identity theories
- 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 t ...

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