Note that our

language L (see

NSA: what's a language? for more details of

*that*) only has

names for M, not for *M. Thus, any

sentence S of the language has an interpretation in both *M and M; we're guaranteed that it will receive the same

truth value in both.

In particular, while speaking in L, the language of M, we can only ever access elements of *M in a sentence by using quantifiers. So while we can deduce (and it is true) that there exists an element of *R (the pseudo- real numbers) which is positive and smaller than any positive real number, this element has no name. Worse, we cannot even express this property ("there exists a positive `x` smaller than any positive real number") in our language L. The transfer principle shows this: if sentence S expressed this property, it would be true in *R, but false in R (there is definitely no such positive real number; half of it would be smaller!), contradicting the transfer principle.

We could conceivably *extend* L to the language *L, which will have names for all objects of *M. A sentence T in the language *L has an interpretation in *M, but if it contains any constants not in L then it is meaningless in M.