Internal Set - Internal Sets in The Ultrapower Construction

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:

    The internal effects of a mutable policy ... poisons the blessings of liberty itself.
    James Madison (1751–1836)

    I would rather have as my patron a host of anonymous citizens digging into their own pockets for the price of a book or a magazine than a small body of enlightened and responsible men administering public funds. I would rather chance my personal vision of truth striking home here and there in the chaos of publication that exists than attempt to filter it through a few sets of official, honorably public-spirited scruples.
    John Updike (b. 1932)

    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)