In [Infinitesimal analysis without the Axiom of Choice](https://doi.org/10.1016/j.apal.2021.102959), 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?



SPOT as a conservative extension of Zermelo–Fraenkel
In plain terms, the conservativity of SPOT over ZF means that if a particular statement S in the language of ZF is provable in SPOT, then ZF can already prove S (with a possibly different proof). Note that ZF does not include the axiom of choice.


More formally, the conservativity of SPOT over ZF is a statement about *formal proofs;* it asserts that for every proof $\pi$ of a statement S from the axioms of SPOT, there is a proof $\pi'$ of S from the axioms of ZF.


A much older conservativity proof was noted by Georg Kreisel in 1956. Kreisel observed (on the basis of Gödel's work on the [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe)) that if S is an *arithmetical statement* (i.e., a first order sentence formulated in the usual language of Peano Arithmetic), and S is provable in ZFC + GCH (where ZFC is ZF plus the axiom of choice, and GCH is the general form of the continuum hypothesis) then S is already provable in ZF alone.


So, by Kreisel's observation, if one manages to prove an arithmetical statement (e.g., Goldbach's conjecture) using a "fancy" proof that uses the axiom of choice and/or the continuum hypothesis, there is another "spartan" proof in ZF alone of the same statement. This [FOM post](https://cs.nyu.edu/pipermail/fom/2015-May/018735.html) of mine provides more detail (and further links).


A vast generalization of Kreisel's observation was proved by Shoenfield in 1961, it is known as the Shoenfield [absoluteness](https://en.wikipedia.org/wiki/Absoluteness) theorem, and it is one of the cornerstones of modern set theory.


