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 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.
Read more about Internal Set: Internal Sets in The Ultrapower Construction, Internal Subsets of The Reals
Famous quotes containing the words internal and/or set:
“If the Revolution has the right to destroy bridges and art monuments whenever necessary, it will stop still less from laying its hand on any tendency in art which, no matter how great its achievement in form, threatens to disintegrate the revolutionary environment or to arouse the internal forces of the Revolution, that is, the proletariat, the peasantry and the intelligentsia, to a hostile opposition to one another. Our standard is, clearly, political, imperative and intolerant.”
—Leon Trotsky (1879–1940)
“Early, as well as late,
Rise with the sun, and set in the same bowers;”
—Henry Vaughan (1622–1695)