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:
“The real essence, the internal qualities, and constitution of even the meanest object, is hid from our view; something there is in every drop of water, every grain of sand, which it is beyond the power of human understanding to fathom or comprehend. But it is evident ... that we are influenced by false principles to that degree as to mistrust our senses, and think we know nothing of those things which we perfectly comprehend.”
—George Berkeley (1685–1753)
“The lady—bearer of this—says she has two sons who want to work. Set them at it, if possible. Wanting to work is so rare a merit, that it should be encouraged.”
—Abraham Lincoln (1809–1865)