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In [[mathematical logic]], in particular in [[model theory]] and [[non-standard analysis]], an '''internal set''' is a set that is a member of a model.

The concept of internal sets is a tool in formulating the [[transfer principle]], which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the [[hyperreal number]]s. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to '''internal sets''' rather than to all sets (note that the term "language" is used in a loose sense in the above).

Edward Nelson's [[internal set theory]] is an axiomatic approach to non-standard analysis (see also Palmgren at [[constructive non-standard analysis]]). Conventional infinitary accounts of non-standard analysis also use the concept of internal sets.
==Internal sets in the ultrapower construction==
Relative to the [[ultrapower]] construction of the [[hyperreal number]]s as equivalence classes of sequences <math>\langle u_n\rangle</math>, an internal subset [''A<sub>n</sub>''] of *R is one defined by a sequence of real sets <math>\langle A_n \rangle</math>, where a hyperreal <math>[u_n]</math> is said to belong to the set <math>[A_n]\subset \; ^*\!{\mathbb R}</math> if and only if the set of indices n such that <math>u_n \in A_n</math>, is a member of the [[ultrafilter]] used in the construction of *R.

More generally, an internal entity is a member of the natural extension of a real entity. Thus, every element of *R is internal; a subset of *R is internal if and only if it is a member of the natural extension <math>{ } ^* \mathcal{P}(\mathbb{R})</math> of the power set <math>\mathcal{P}(\mathbb{R})</math> of R; etc.
==Internal subsets of the reals==
Every internal subset of <math>\mathbb{R}</math> is necessarily ''finite'', (see Theorem 3.9.1 Goldblatt, 1998). In other words, every internal infinite subset of the hyperreals necessarily contains non-standard elements.

==See also==
*[[Standard part function]]

== References ==
*[[Robert Goldblatt|Goldblatt, Robert]]. ''Lectures on the [[hyperreal]]s''. An introduction to nonstandard analysis. [[Graduate Texts in Mathematics]], 188. Springer-Verlag, New York, 1998.

* {{citation | title=Non-standard analysis | author=Abraham Robinson | authorlink=Abraham Robinson | series=Princeton landmarks in mathematics and physics | publisher=Princeton University Press | year=1996 | isbn=978-0-691-04490-3 }}

{{Infinitesimals}}

[[Category:Non-standard analysis]]

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