mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, 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 (math ...

and

nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

, 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 In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...

, which concerns the logical relation between the properties of the

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

numbers R, and the properties of a larger field denoted *R called the

hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...

s. The field *R includes, in particular,

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

("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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

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 Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematic ...

internal set theory Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, ...

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 The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...

construction of the

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 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 filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...

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

hyperreal Hyperreal may refer to: * Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis * Hyperreal.org, a rave culture website based in San Francisco, US * Hyperreality, a term used in semiotics and po ...

s''. An introduction to nonstandard analysis.

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) 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 ...

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

Internal sets in the ultrapower construction

Internal subsets of the reals

See also

References