  Reverse Mathematics (RM) is a program in the foundations of mathematics
founded by Friedman and developed extensively by Simpson. The aim of RM is
finding the minimal axioms needed to prove a theorem of ordinary (i.e. non-set
theoretical) mathematics. In the majority of cases, one also obtains an
equivalence between the theorem and its minimal axioms. This equivalence is
established in a weak logical system called the base theory; four prominent
axioms which boast lots of such equivalences are dubbed mathematically natural
by Simpson. In this paper, we show that a number of axioms from Nonstandard
Analysis are equivalent to theorems of ordinary mathematics not involving
Nonstandard Analysis. These equivalences are proved in a weak base theory
recently introduced by van den Berg and the author. In particular, our base
theories have the first-order strength of elementary function arithmetic, in
contrast to the original version of this paper [22]. Our results combined with
Simpson's criterion for naturalness suggest the controversial point that
Nonstandard Analysis is actually mathematically natural.
