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, (see Theorem 3.9.1 Goldblatt, 1998). In other words, every internal infinite subset of the hyperreals necessarily contains non-standard elements.

See also

• Standard part function
• Superstructure (mathematics)

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