# Internal set

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{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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 numbers. 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 numbers as equivalence classes of sequences ${\displaystyle \langle u_{n}\rangle }$, an internal subset [An] of *R is one defined by a sequence of real sets ${\displaystyle \langle A_{n}\rangle }$, where a hyperreal ${\displaystyle [u_{n}]}$ is said to belong to the set ${\displaystyle [A_{n}]\subset \;^{*}\!{\mathbb {R} }}$ if and only if the set of indices n such that ${\displaystyle u_{n}\in A_{n}}$, 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 ${\displaystyle {}^{*}{\mathcal {P}}({\mathbb {R} })}$ of the power set ${\displaystyle {\mathcal {P}}({\mathbb {R} })}$ of R; etc.

## Internal subsets of the reals

Every internal subset of ${\displaystyle \mathbb {R} }$ 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.

## References

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