Internal set

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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 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[edit]

Relative to the ultrapower construction of the hyperreal numbers as equivalence classes of sequences \langle u_n\rangle, an internal subset [An] of *R is one defined by a sequence of real sets \langle A_n \rangle, where a hyperreal [u_n] is said to belong to the set [A_n]\subset \; ^*\!{\mathbb R} if and only if the set of indices n such that 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 { } ^* \mathcal{P}(\mathbb{R}) of the power set \mathcal{P}(\mathbb{R}) of R; etc.

Internal subsets of the reals[edit]

Every internal subset of \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.

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