珍贵数学文献（II）

   荷兰千年名校格罗宁根大学知名数学J.Ponstein教授为不的熟悉数理逻辑的数学工作者精心撰写了一部介绍“非标准分析”科普专著，对于我国高校微积分教学改革很有帮助。

为此，我们将此文分为两个部分顺序转发。请见本文附件。

对此专著的分析与评论随后再发、

袁萌  陈启清  9月28日

附件：Non-standard analysis

Cnapter  3

Some applications

3.1 Introduction and least upper bound theorem

The aim of this chapter is to show how many denitions and proofs of elementary calculus can be simplied by means of nonstandard analysis. Only a number of important examples will be considered. A much more complete treatment is Keisler [26], where the existence of nonstandard numbers is taken for granted, however, and a simplied form of transfer is introduced in an axiomatic kind of way.

Theorem 3.1.1 (The least upper bound theorem.) Let S be a nonempty subset of IR that is bounded above by some (classical) real number. Then S has a least upper bound in IR.

Proof:

Taking any c ∈ S, instead of S we may consider {s : s ∈ S, s ≥ c}, that is to say we may assume that s ≥ c for all s ∈ S. Then c, b ∈ IR, c < b, exist such that ∀s ∈ S : c ≤ s ≤ b, so that, by transfer, ∀s ∈∗ S : c ≤ s ≤ b. Let ω ∈∗ IN, ω ∼ ∞ be arbitrary and divide ∗[c,b] in ω equal subintervals of length δ = (b−c)/ω, so that δ ∼ 0, and consider the points a, a + δ, a + 2δ, ..., a + ωδ = b. Then, ∃j ∈∗IN : [∀s ∈∗S : s ≤ a + jδ] ∧ [∃s0 ∈∗S : s0 > a + jδ−δ].

Let β =st(a+jδ), which is well dened as a+jδ is limited. Then β is a (hence the) least upper bound of S. For rst of all if s ∈ S then s ∈∗S, hence s ≤ a+jδ = β+ε for some ε ' 0, but since s, β ∈ IR this means that s ≤ β. And secondly, if β0 were a smaller upper bound of S, then β > β0 + 1/m for some m ∈ IN, hence

101

102