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  "body": "\n## Summary\n\nInternal set theory, written IST, is an axiomatic form of [[nonstandard analysis|WM62GKCBWN]] introduced by Edward Nelson in 1977.\nIt adds to [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice a new unary predicate \"standard\" and three axiom schemes: idealization, standardization and transfer.\nInstead of building a larger number system, IST finds nonstandard elements, such as infinitesimal reals and unlimited natural numbers, inside the usual sets.\nNelson proved that IST is a conservative extension of ZFC.\nSo every statement in the language of ZFC that IST proves is already a theorem of ZFC.\n\n## Explanations\n\n### L1 human\n\nIn IST the real line is the usual real line, but some of its elements are labelled \"standard\" and others \"nonstandard\".\nEvery number you can name explicitly, such as $0$, $1$ or $\\pi$, is standard.\nThe axioms guarantee that there are also nonstandard numbers, for example a natural number larger than every standard one.\nIts reciprocal is then a positive real smaller than every standard positive real, an infinitesimal.\nOne may use these numbers in proofs, but one may not form \"the set of all standard numbers\".\n",
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