  We survey the effective foundations for analysis with infinitesimals
developed by Hrbacek and Katz in 2021, and detail some applications. Theories
SPOT and SCOT are conservative over respectively ZF and ZF+ADC. The range of
applications of these theories illustrates the fact that analysis with
infinitesimals requires no more choice than traditional analysis. The theory
SCOT incorporates in particular all the axioms of Nelson's Radically Elementary
Probability Theory, which is therefore conservative over ZF+ADC.
