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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 ...
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 ...
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.
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 ...
A structure which has a pair of non-zero elements, ... The completion with respect to the usual absolute value (from the order) is the field of real numbers.
The rational numbers form an initial totally ordered set which is dense in the real numbers. Moreover, the reflexive reduction < is a dense order on the rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are totally ordered by definition. They ...
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 ...
For instance, the function that maps a natural number to its successor is clearly monotone with respect to the natural order. Any function from a discrete order, i.e. from a set ordered by the identity order "=", is also monotone. Mapping each natural number to the corresponding real number gives an example for an order embedding.