mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, in particular in

model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

and nonstandard 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 number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer

n

s. The field *R includes, in particular,

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

("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 nonstandard analysis (see also Palmgren at constructive nonstandard analysis). Conventional infinitary accounts of nonstandard analysis also use the concept of internal sets.

Internal sets in the ultrapower construction

Relative to the ultrapower construction of the

n

s as equivalence classes of sequences

\langle u_n\rangle

of reals, an internal subset 'A_n''of *R is one defined by a sequence of real sets

\langle A_n \rangle

, where a hyperreal

_n /math> is said to belong to the set_n subseteq \; ^*\! if and only if the set of indices ''n'' such that u_n \in A_n, is a member of the

ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

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(\mathbb)

of the power set

\mathcal(\mathbb)

of R; etc.

Internal subsets of the reals

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

References

* Goldblatt, Robert. ''Lectures on the hyperreals''. An introduction to nonstandard analysis.

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...

, 188. Springer-Verlag, New York, 1998. * {{Infinitesimals Nonstandard analysis

Internal sets in the ultrapower construction

Internal subsets of the reals

See also

References