Key Application of the Nonstandard Methods

In [14, Chap. 2] the construction of the nonstandard universe is used twice: first to obtain R, the field of real numbers, from the field Q of the rationals; on second application, to work out the structure of *R from R. The first use can supersede such classical constructions as the ones devised by George Cantor and Richard Dedekind.

The second use brings infinitesimals into play, along with their inverses, which are infinite numbers: one is thus led into the realm of hyperreal numbers.

To briefly see how these embeddings work, consider first an ordered field D (in the customary sense). For any such field, we can assume w.l.o.g. that Q c D.

Definition 10.5 Put

An element x of D is said to be finite, infinite, or infinitesimal, depending as whether x e F, x e D F, or x e I. For x, y in D, we say that x is near y if x - y e I; if so, we write x & y.

D is called Archimedean if F = D; otherwise stated, if I = {0}. 4

As is plain, I is an ideal in the subring F of D; moreover, & is an equivalence relation on D, whose restriction to F equals the equivalence relation induced by

I. Consequently, the quotient F/ &= F/1 is a ring; actually, it is an Archimedean ordered field.

Suppose next that D is an Archimedean ordered field and that D c s, where s is as in Sects. 10.2, 10.3, and 10.4. By virtue of the transfer principle, the *D resulting from D through the ultrapower construction is, in its turn, an ordered field (of which D is a subfield). It is no longer Archimedean, though; for, its nonnull subset IN s consists of elements which are infinite. If we now designate by F and I the set of all finite, respectively infinitesimal, elements of +D, then it readily turns out that the canonical homomorphism ° of F onto F/I acts as a monomorphism of D into F/I. After so embedding D in the Archimedean field F/I, [14, p. 51] goes on to prove that F, D, I, and 1D F are all external subsets of 1D; then, by resorting to the concurrence theorem, [14] obtains the following:

Theorem 10.7 (Dedekind’s Theorem) If A, B are nonnull subsets of D such that a < b holds for all a e A and b e B, then there is a c e F/I such that a < c < b holds for all a e A and b e B.

From this, [14] gets that

Theorem 10.8 F/I is a complete ordered field,

after noting that between two elements x, y of an Archimedean ordered field such that x < y there always lies a q e Q such that x < q < y. Archimedean ordered fields exist (one such is, of course, Q); therefore, a complete ordered field exists as well. Up to isomorphism, this must be unique (owing, in particular, to the fact that any complete ordered field is Archimedean): by definition, R is taken to be this field.

If we go over the same construction again, now taking D = R c s, we can naturally identify F/I with R and, accordingly, think of ° as being the field homomorphism that sends each finite hyperreal number to its standard part, namely to the sole real number which lies near it. It can also be shown (see [14, pp. 53, 56]) that infinitesimally near each real number there is a q e *Q.

Typical notions of elementary real analysis can be captured in new terms from the nonstandard viewpoint, after which classical theorems can be obtained by nonstandard methods. Various illustrations of this are provided in [14, pp. 56-74], e.g.:

Theorem 10.9 Consider a sequence {sn : n e N {0}} of real numbers sn and a real number l. Then

  • the sequence converges to l if and only if fs )n & l holds for all infinite n e IN;
  • (*s)n & l holds for some infinite n e IN if and only if, for each e > 0 in R, the inequality sn l| < e is satisfied for infinitely many n e N.

Theorem 10.10 Let f be a real-valued function on the closed interval [a, b] =Def {x e R a < x < b}, where a, b e R and a < b. Then f is continuous at x0 e [a, b] if and only if, for all x e *[a, b], x & x0 implies *f (x) & *f (x0).

Theorem 10.11 Let f be a continuous real-valued function on the closed interval [a, b]. If f (a) < 0 < f (b), then f (c) = 0 holds for some c e [a, b].

Proof 1 (Sketch) Consider the function t : N x N —> R defined as follows:

so that *t : IN x IN —> 1R meets an analogous condition, by the transfer principle.

Choose v e IN N. Since L = {i e IN f (*t(v, i)) > 0 and i < v} is a definable subset of *s, L is also internal by Theorem 10.4; and since v e L, there is a least element j > 0inL .Ifwetakec to be the standard part of *t (v, j), it turns out that c & *t(v, j) & *t(v, j — 1); therefore f (c) & f (ft(v, j)) & f (*t(v, j — 1)), and hence f (c) = o( f (*t(v, j))) = o(f (*t(v, j — 1))), where the inequalities °(f (*t(v, j))) > 0 and °(f (*t(v, j — 1))) < 0 hold. We conclude that f (c) = 0, as desired. ?

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