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
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