Mathematics Logic

# Logic

List of large cardinal properties
Main article: Large cardinalThis page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of c ...
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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 ...
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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 ...
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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 ...
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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∀ ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ¬ ...
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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 ∀ ...
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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 ...
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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 ...
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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; ...
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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 ...
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