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Nonstandard Models the atoms in ∗ S = σ S, we may assume that ∗ S = S and that ∗ : S → ∗ S is the identity. Since σ A = ∗ A for all sets A, we may conclude that ∗ : S → ∗ S is the identity. 12 (Standard Definition Principle for Relations). An n-ary relation A ∈ ∗ S is standard if and only if it can be written in the form A = {(x1 , . . , xn ) ∈ B1 × · · · × Bn : ∗ α(x1 , . . , xn )} where ∗ α is a transitively bounded predicate with x1 , . . , xn as its only free variables, and B and all elements (=constants) occurring in ∗ α are standard elements.

The restriction of an internal function to an internal set is internal. From the above observations one might guess that I is the same as ∗ S, because many “natural” operations appear to remain within the nonstandard universe I . In fact, the earlier mentioned approach to nonstandard analysis by Nelson only “knows” internal sets: This approach is more or less an axiomatic description of set theory within I , the so-called internal set theory (this is not quite precise, but gives a rather good idea of Nelson’s approach).

1 Ultrafilters Let J be some set. Probably, the reader has already heard the notion that a property holds “almost everywhere” on J: By this, one means that the set of all point j ∈ J with this property is “large” in a certain specified sense. For example, one may mean that the complement of this set is finite (if J is infinite); if J is a measure space, one can also mean that the complement of this set is a null set. (Recall Exercises 3 and 4). If we want to introduce a general definition of the term “almost everywhere” which contains the two cases above, we should fix a family F of subsets of J and say that a property holds almost everywhere if the set of all points j ∈ J with this property is an element of F .