Real closed field in the context of "Hyperreal number"

In mathematics, the hyperreal numbers, denoted ${\displaystyle \ast \mathbb {R} }$, are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number ${\displaystyle x}$ is said to be finite when ${\displaystyle |x|<n}$ for some integer ${\displaystyle n}$. Similarly, ${\displaystyle x}$ is said to be infinitesimal when ${\displaystyle |x|<1/n}$ for all positive integers ${\displaystyle n}$. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about ${\displaystyle \mathbb {R} }$ are also valid in ${\displaystyle *\mathbb {R} }$. For example, the commutative law of addition, ${\displaystyle x+y=y+x}$, holds for the hyperreals just as it does for the reals; since ${\displaystyle \mathbb {R} }$ is a real closed field, so is ${\displaystyle *\mathbb {R} }$. Similarly, since ${\displaystyle \sin({\pi n})=0}$ for all integers ${\displaystyle n}$, one also has ${\displaystyle \sin({\pi H})=0}$ for all hyperintegers ${\displaystyle H}$. The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955.