Editing Internal set theory (section)

==Formal justification for the axioms==
Aside from the intuitive motivations suggested above, it is necessary to justify that additional IST axioms do not lead to errors or inconsistencies in reasoning. Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of [[Gottfried Leibniz]], [[Johann Bernoulli]], [[Leonhard Euler]], [[Augustin-Louis Cauchy]], and others were the reason that they were originally abandoned for the more cumbersome<ref>{{cite book |last1=Vopěnka |first1=Petr |title=Mathematics in the Alternative Set Theory (Teubner 1979).pdf |date=1979 |publisher=BSB B. G. Teubner Verlagsgesellschaft |location=Leipzig |url=https://drive.google.com/file/d/17JRj2orUVDw7lrBEmBS1K6OK06RP32Xa/view |access-date=3 April 2025 |ref=VopenkaTeubner}}</ref> [[real number]]-based arguments  developed by [[Georg Cantor]], [[Richard Dedekind]], and [[Karl Weierstrass]], which were perceived as being more rigorous by Weierstrass's followers.

The approach for internal set theory is the same as that for any new axiomatic system—we construct a [[model theory|model]] for the new axioms using the elements of a simpler, more trusted, axiom scheme. This is quite similar to justifying the consistency of the axioms of [[Elliptic geometry|elliptic]] [[non-Euclidean geometry]] by noting they can be modeled by an appropriate interpretation of [[great circle]]s on a sphere in ordinary 3-space.

In fact via a suitable model a proof can be given of the relative consistency of IST as compared with ZFC: if ZFC is consistent, then IST is consistent.  In fact, a stronger statement can be made: IST is a [[conservative extension]] of ZFC: any internal formula that can be proven within internal set theory can be proven in the Zermelo–Fraenkel axioms with the axiom of choice alone.<ref>Nelson, Edward (1977). Internal set theory: A new approach to nonstandard analysis. [[Bulletin of the American Mathematical Society]] 83(6):1165–1198.</ref>