TITLE: SPOT as a conservative extension of Zermelo–Fraenkel
QUESTION [8 upvotes]: In Infinitesimal analysis without the Axiom of Choice, Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make use of infinitesimals (known as SPOT, an acronym of its axioms).
Elsewhere in another somewhat related article by Katz I have read that SPOT is conservative over traditional Zermelo–Fraenkel set theory and so does not depend on the axiom of choice or on the existence of ultra-filters.  Can someone explain in terms more suitable for a non-expert what it means for SPOT to be “conservative” over ZF, and also why this implies no dependence on the axiom of choice?

REPLY [11 votes]: Ali's answer contains all the required technical explanation.  My answer is more "sociological background " in nature:
For various reasons, there have been rather strong negative reactions to Nonstandard Analysis.  Bishop and Connes come to mind as good examples, as discussed at length in papers by M. Katz and his co-authors.
By the usual standard of mathematics, the negative responses to Nonstandard Analysis can only be called "fierce" and "highly emotional".  To dress up this semi-religious war with intellectual arguments, one often hears that
"Nonstandard Analysis more heavily makes use of the Axiom of Choice (than usual mathematics)."
The point of SPOT is to show that this commonplace is actually false.