Internal set

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 $\langle u_{n}\rangle$ , an internal subset [A_n] 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

Every internal subset of $\mathbb {R}$ is necessarily finite, (e.g. has no infinite elements, but can have infinitely many elements; see Theorem 3.9.1 Goldblatt, 1998). In other words, every internal infinite subset of the hyperreals necessarily contains non-standard elements.

References

Goldblatt, Robert. Lectures on the hyperreals. 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

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Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Method of indivisibles
Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive non-standard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity
Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

Internal set

Internal sets in the ultrapower construction

Internal subsets of the reals

See also

References