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The set of real numbers has several standard structures: An order: each number is either less than or greater than any other number. Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
Tarski's axiomatization, which is a second-order theory, can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered field; it is however made much more concise by avoiding multiplication altogether and using unorthodox variants of standard algebraic axioms and other subtle tricks. Tarski did ...
On the other hand, the completions with respect to the other non-trivial absolute values give the fields of p-adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the ultrametric property, then the -adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).
For example, the real numbers form an Archimedean field, but hyperreal numbers form a non-Archimedean field, because it extends real numbers with elements greater than any standard natural number. [4] An ordered field F is isomorphic to the real number field R if and only if every non-empty subset of F with an upper bound in F has a least upper ...
The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound.
Mapping each natural number to the corresponding real number gives an example for an order embedding. The set complement on a powerset is an example of an antitone function. An important question is when two orders are "essentially equal", i.e. when they are the same up to renaming of elements.