Internal Sets in The Ultrapower Construction
Relative to the ultrapower construction of the hyperreal numbers as equivalence classes of sequences, an internal subset 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.
Read more about this topic: Internal Set
Famous quotes containing the words internal, sets and/or construction:
“A State, in idea, is the opposite of a Church. A State regards classes, and not individuals; and it estimates classes, not by internal merit, but external accidents, as property, birth, etc. But a church does the reverse of this, and disregards all external accidents, and looks at men as individual persons, allowing no gradations of ranks, but such as greater or less wisdom, learning, and holiness ought to confer. A Church is, therefore, in idea, the only pure democracy.”
—Samuel Taylor Coleridge (1772–1834)
“A continual feast of commendation is only to be obtained by merit or by wealth: many are therefore obliged to content themselves with single morsels, and recompense the infrequency of their enjoyment by excess and riot, whenever fortune sets the banquet before them.”
—Samuel Johnson (1709–1784)
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)