We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of G-sets and Boolean étendues.
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José Siqueira (2024) studied this question.
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