Real closed field

In mathematics, a real closed field is a field ${\displaystyle F}$ that has the same first-order properties as the field of real numbers. (First-order properties are those properties that can be expressed with the logic symbols ${\displaystyle \forall ,\exists ,\vee ,\land ,\neg ,\to }$ and the arithmetic symbols ${\displaystyle 0,1,+,-,\times ,\div ,=}$, where the domain of all quantifiers is the set ${\displaystyle F}$; it is hence not allowed to quantify over natural numbers, subsets of ${\displaystyle F}$, sequences in ${\displaystyle F}$, functions ${\displaystyle F\to F}$ etc.) Some examples of real closed fields are the field of real numbers itself, the field of real algebraic numbers, and fields of hyperreal numbers that include infinitesimals. In algebra, most theorems that involve the real numbers remain true when formulated for arbitrary real closed fields.