Internal set
Encyclopedia
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.
Internal set is the key 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 hyperreals. 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 not a constructivist
version of non-standard analysis (but see Palmgren at constructive non-standard analysis
). Its name should not mislead the reader: conventional infinitary accounts of non-standard analysis also use the concept of internal sets.
s as equivalence classes of sequences
, an internal subset [An] 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.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, in particular in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, an internal set is a set that is a member of a model.
Internal set is the key tool in formulating the transfer principle
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...
, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreals. 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
Internal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
is not a constructivist
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
version of non-standard analysis (but see Palmgren at constructive non-standard analysis
Constructive non-standard analysis
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's non-standard analysis, developed by Moerdijk , Palmgren , Ruokolainen . Ruokolainen wrote:-References:...
). Its name should not mislead the reader: 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 numberHyperreal number
The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s as equivalence classes of sequences
, an internal subset [An] 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 ultrafilterUltrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
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.