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.
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Famous quotes containing the words internal, sets and/or construction:
“What makes some internal feature of a thing a representation could only its role in regulating the behavior of an intentional system.”
—Daniel Clement Dennett (b. 1942)
“Wilson adventured for the whole of the human race. Not as a servant, but as a champion. So pure was this motive, so unflecked with anything that his worst enemies could find, except the mildest and most excusable, a personal vanity, practically the minimum to be human, that in a sense his adventure is that of humanity itself. In Wilson, the whole of mankind breaks camp, sets out from home and wrestles with the universe and its gods.”
—William Bolitho (1890–1930)
“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)