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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== Relative to the ultrapower construction of the hyperreal numbers as equivalence classes of sequences , an internal subset () of *R is one defined by a sequence of real sets , where a hyperreal is said to belong to the set if and only if the set of indices n such that , 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 of the power set of R; etc. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「internal set」の詳細全文を読む スポンサード リンク
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