Given R* as defined, we know that R ̹ R* because we have a Î R* such that the interpretation of P_<(r, v₁) in R* is true for v₁ = a for any standard r; i.e. r* <* a.

So a is an infinitely large number.

So 1/a (if it is defined) is an infinitely small number.

Definitions

1.    F = {x Î R*: |x|* <* y for some y Î R}               

        The finite elements of | R* |

2.    I = {x Î R*: |x|* <* y for all y Î R⁺}               

        The infinitesimals of | R* |

.    There are many infinitely large numbers.

If A Ì R is unbounded then the sentence  ("r)($a Î A) a > r is true in R, and also in R*. So if a is an infinite number in A* then there is a more infinite number a' also in A*.

.    There are many infinitely small numbers

For each infinite a there is an infinitesimal 1/*a in R*.

Note that in  R there is just one infinitesimal, i.e. 0.

Properties

1.    F is closed under +*, -*, .*

       If x, y, are finite then $a, b Î R | x |* <* a, | y |* <* b

        | x ±* y |* £* | x |* + | y |* < a + b

        | x .* y |* < a + b

2.    I is closed under +*, -*, and multiplication by finites.

       If x, y, are infinitesimals then "a Î R* | x |* <* a/2, | y |* <* a/2

       | x ±* y |* <* a/2 + a/2

       If z is finite then $b Î R | z |* <* b

       Now | x |* <* a/b so |x .* y |* <* (a/b)a = a