In the book Radically Elementary Probability Theory, Nelson presented an elementary nonstandard axiomatic system and developed an alternative approach to advanced stochastics, avoiding measure theory. This axiomatic system resembles Peano Arithmetic and assigns the predicate “standard” only to natural numbers. The present paper aims to incorporate standard real numbers in this approach, while remaining reasonably elementary. The real numbers will be external sets within a fragment of Kanovei and Reeken’s system \(\mathrm {HST}\) , which is based on a bounded form of Nelson’s Internal Set Theory; however, we avoid the axiom schemes of the latter. We thus obtain a real closed field which enables an interplay between nonstandard analysis and classical analysis for definable properties. We point out how this structure relates to the standard real numbers of Internal Set Theory, and to Bolzano’s distinction between elementary numbers and measurement numbers.