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:
“We have our difficulties, true; but we are a wiser and a tougher nation than we were in 1932. Never have there been six years of such far flung internal preparedness in all of history. And this has been done without any dictator’s power to command, without conscription of labor or confiscation of capital, without concentration camps and without a scratch on freedom of speech, freedom of the press or the rest of the Bill of Rights.”
—Franklin D. Roosevelt (1882–1945)
“Analysis as an instrument of enlightenment and civilization is good, in so far as it shatters absurd convictions, acts as a solvent upon natural prejudices, and undermines authority; good, in other words, in that it sets free, refines, humanizes, makes slaves ripe for freedom. But it is bad, very bad, in so far as it stands in the way of action, cannot shape the vital forces, maims life at its roots. Analysis can be a very unappetizing affair, as much so as death.”
—Thomas Mann (1875–1955)
“There’s no art
To find the mind’s construction in the face:
He was a gentleman on whom I built
An absolute trust.”
—William Shakespeare (1564–1616)