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Inmathematical logic, in particular inmodel theory andnonstandard analysis, aninternal set is a set that is a member of a model.

The concept of internal sets is a tool in formulating thetransfer principle, which concerns the logical relation between the properties of thereal numbersR, and the properties of a largerfield denoted *R called thehyperreal 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 expressanalysis overR 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 theset-theoretic level, the propositions in such a language are interpreted to apply only tointernal sets rather than to all sets (note that the term "language" is used in a loose sense in the above).

Edward Nelson'sinternal set theory is an axiomatic approach to nonstandard analysis (see also Palmgren atconstructive nonstandard analysis). Conventional infinitary accounts of nonstandard analysis also use the concept of internal sets.

Relative to theultrapower construction of thehyperreal numbers as equivalence classes of sequences${\displaystyle ⟨u_n⟩}$ of reals, an internal subset [A_n] of *R is one defined by a sequence of real sets${\displaystyle ⟨A_n⟩}$, where a hyperreal${\displaystyle [u_n]}$ is said to belong to the set${\displaystyle [A_n]\subseteq {}^{\ast }\mathbb{R}}$ if and only if the set of indicesn such that${\displaystyle u_n\in A_n}$, is a member of theultrafilter 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${\displaystyle {}^{\ast }\mathcal{P}(\mathbb{R})}$ of the power set${\displaystyle \mathcal{P}(\mathbb{R})}$ ofR; etc.

Every internal subset of *R that is a subset of (the embedded copy of)R is necessarilyfinite (see Theorem 3.9.1 Goldblatt, 1998). In other words, every internal infinite subset of the hyperreals necessarily contains nonstandard elements.

• Standard part function
• Superstructure (mathematics)

• Goldblatt, Robert.Lectures on thehyperreals. An introduction to nonstandard analysis.Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998.
• Abraham Robinson (1996),Non-standard analysis, Princeton landmarks in mathematics and physics, Princeton University Press,ISBN 978-0-691-04490-3

• Nonstandard analysis

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