Non-standard Analysis - Internal Sets

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, ..., x_n) with bounded quantification and with free variables x₁, ..., x_n, and for any elements A₁, ..., A_n of V(R), the following equivalence holds:

• Countable saturation: If {A_k}_{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.