Internal Sets
A set x is internal if and only if x is an element of *A for some element A of V(R). *A itself is internal if A belongs to V(R).
We now formulate the basic logical framework of nonstandard analysis:
- Extension principle: The mapping * is the identity on R.
- Transfer principle: For any formula P(x1, ..., xn) with bounded quantification and with free variables x1, ..., xn, and for any elements A1, ..., An of V(R), the following equivalence holds:
- Countable saturation: If {Ak}k ∈ N is a decreasing sequence of nonempty internal sets, with k ranging over the natural numbers, then
One can show using ultraproducts that such a map * exists. Elements of V(R) are called standard. Elements of *R are called hyperreal numbers.
Read more about this topic: Non-standard Analysis
Famous quotes containing the words internal and/or sets:
“We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
—Max Lerner (b. 1902)
“The vain man does not wish so much to be prominent as to feel himself prominent; he therefore disdains none of the expedients for self-deception and self-outwitting. It is not the opinion of others that he sets his heart on, but his opinion of their opinion.”
—Friedrich Nietzsche (1844–1900)