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:
“The internal effects of a mutable policy ... poisons the blessings of liberty itself.”
—James Madison (1751–1836)
“To me this world is all one continued vision of fancy or imagination, and I feel flattered when I am told so. What is it sets Homer, Virgil and Milton in so high a rank of art? Why is Bible more entertaining and instructive than any other book? Is it not because they are addressed to the imagination, which is spiritual sensation, and but mediately to the understanding or reason?”
—William Blake (1757–1827)