Internal set

In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, in particular in <a href="/facts/Model_theory/vLWVV8sQ">model theory</a> and <a href="/facts/Nonstandard_analysis/lEpNpWMB">nonstandard analysis</a>, an internal set is a set that is a member of a model.
The concept of internal sets is a tool in formulating the <a href="/facts/Transfer_principle/1lv1hA3h">transfer principle</a>, which concerns the logical relation between the properties of the <a href="/facts/Real_number/R02gw5Pb">real</a> numbers R, and the properties of a larger <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> denoted *R called the <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreal numbers</a>. The field *R includes, in particular, <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express <a href="/facts/Real_analysis/12WjlXCW">analysis over R</a> 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 <a href="/facts/Set-theoretic/1ptfsvUP">set-theoretic</a> 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).
<a href="/facts/Edward_Nelson/mn8Yo691">Edward Nelson</a>'s <a href="/facts/Internal_set_theory/RLpCIhxQ">internal set theory</a> is an axiomatic approach to nonstandard analysis (see also Palmgren at <a href="/facts/Constructive_nonstandard_analysis/SJx3wjoY">constructive nonstandard analysis</a>). Conventional infinitary accounts of nonstandard analysis also use the concept of internal sets.

Internal set open-in-new

Internal set