---
tag: V0PGBXYJ9R
kind: concept
name: Internal set theory
msc: ["03H05", "03E70", "26E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Internal_set_theory
same_as:
  wikidata: Q1666284
  wikipedia: Internal set theory
---

## Summary

Internal set theory, written IST, is an axiomatic form of [[nonstandard analysis|WM62GKCBWN]] introduced by Edward Nelson in 1977.
It adds to [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice a new unary predicate "standard" and three axiom schemes: idealization, standardization and transfer.
Instead of building a larger number system, IST finds nonstandard elements, such as infinitesimal reals and unlimited natural numbers, inside the usual sets.
Nelson proved that IST is a conservative extension of ZFC.
So every statement in the language of ZFC that IST proves is already a theorem of ZFC.

## Explanations

### L1 human

In IST the real line is the usual real line, but some of its elements are labelled "standard" and others "nonstandard".
Every number you can name explicitly, such as $0$, $1$ or $\pi$, is standard.
The axioms guarantee that there are also nonstandard numbers, for example a natural number larger than every standard one.
Its reciprocal is then a positive real smaller than every standard positive real, an infinitesimal.
One may use these numbers in proofs, but one may not form "the set of all standard numbers".