This

lemma presents an

attractive packaging to the

weak saturation condition which appears in

NSA: Introduction and Construction, in the setting of an

enlargement of

ZF set theory.

So let M be a model of ZF set theory, and let *M be its enlargement (the extension described in NSA: Introduction and Construction). L, the language of M, has a constant name, say s, for *every set* of M. *M also assigns an interpretation to L, so in particular it assigns some object to s.

#### Lemma.

Let s be a name of a set in M. Then s *contains

non-standard *elements in *M

iff s is

infinite in M.

For any element a of s in M, we may form the sentence "a is an element of s" in L, and by the

transfer principle a is an element of s in *M, too; applying the same principle to a's which are

*not* elements of s in M, we see that the

*only* **standard** *elements of s in *M are those which are elements of s in M.

What about non-standard elements?

- Suppose s is finite in M. Enumerate (the names of) its members as a
_{1},...,a_{k}. Now consider this sentence S of L: "for all `x` in s, `x`=a_{1} or ... or `x`=a_{k}" (note that this sentence can be very long, when `k` is very large; but we can always write it down once and for all, enumerating each element a_{j} of s; this is where we use the finiteness of s).
The sentence S is true in M, therefore it's true in *M. Thus we conclude that these `k` elements are the *only* elements of s in *M, too -- s *contains no non-standard elements.

- Now suppose s in infinite. For every a in s, create the formula
`f`^{a}(`x`) "`x` is an element of s, and `x` != a". Let F be the set of all such `f`^{a}. Since s is infinite, F is finitely satisfiable (for instance, to satisfy {`f`^{a1},`f`^{a2},`f`^{a3}}, pick `x` to be an element of s (in M) other than a_{1},a_{2},a_{3}; this is possible since s is infinite, and therefore contains more than 3 elements).
By the weak saturation condition, F is satisfiable in *M. That is, there exists some object *p of *M, such that for every element a in s (in M!) *p != a, and yet *p is a *element of s (in *M). In other words, *p is non-standard (hence has no name in L; if it had a name in L, it would have been in s in M too, and therefore been included in F), and a *element of s in *M. Thus s *contains non-standard *elements.