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 not a constructivist version of non-standard analysis (but see Palmgren at constructive non-standard analysis). Its name should not mislead the reader: conventional infinitary accounts of non-standard analysis also use the concept of internal sets.
Read more about Internal Set: Internal Sets in The Ultrapower Construction, Internal Subsets of The Reals
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—James Madison (1751–1836)
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