Real closed field
In mathematics, a real closed field is a field F {\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 0 , 1 , + , − , × , ÷ , = {\displaystyle 0,1,+,-,\times ,\div ,=} , where the domain of all quantifiers is the set F {\displaystyle F} ; it is hence not allowed to quantify over natural numbers, subsets of F {\displaystyle F} , sequences in F {\displaystyle F} , functions F → F {\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.