1.5 数学鲜花遍地开，课程改革微积分

今年国庆期间，数学鲜花遍地开，课程改革微积分，国内学界同庆。

六十年来，*变换是数学的一朵鲜花。请见本文附件。

袁萌陈启清 10月8日

附件：

1.5 More about the ∗-transform; transfer So far a number of isolated instances of ∗-transforms have been presented (the ∗-transform of IR and other sets, of functions from IR to IR and of statement (1.1) in the preceding section). Although it is too early to present a complete treatment of the ∗-transform, a number of interesting aspects of this notion may be discussed already now. To each number, each set, each function, each operation (such as + and ∪), each simple relation (such as < and ∈), each logical connective (¬, ∨, ∧, ⇒, ⇔), both quantiﬁers (∀, ∃), each deﬁnition, and each statement of classical mathematics, there corresponds a unique ∗-transform in nonstandard mathematics.

The notation is quite simple: just add an asterisk to the upper left of the symbol representing what is to be transformed. Sometimes the ∗-transform is identical to its inverse image, but often this is not so. In the former case the asterisk should, of course, be dropped, but even in the latter case this can sometimes be done without creating confusion.

Below a number of typical examples is presented, but full details will only be given later on. a) Numbers. If x ∈ IR, then ∗x = x. b) Sets. If X is a ﬁnite set of numbers, then ∗X = X, and also (happily so) ∗∅ = ∅, but if X is an inﬁnite set of numbers, then X is strictly included in ∗X (in case X is an arbitrary abstract set and ∗X 6= X, X need not be a subset of ∗X.) c) Pairs. If hx,yi is a pair, then ∗hx,yi = hx∗,y∗i, and similarly for n-tuples hx1,...,xni.

d) Functions. If f : X → Y , then ∗f : ∗X → ∗Y , and ∗f(x) = f(x) if x ∈ X. Often the asterisk in ∗f may be dropped. e) Operations. As an example consider addition in IR. Its ∗-transform is ∗addition in ∗IR, and x∗+y = x + y if x,y ∈ IR. The asterisk can safely be dropped. f) Atomic relations. These are relations in which neither logical connectives nor quantiﬁers play a part, but only such relations as < or∈, etc. Consider ﬁrst < in IR, leading to ∗ < in ∗IR. Similarly as under e) we have that x ∗< y is equivalent to x < y if x,y ∈ IR, and again the asterisk can safely be dropped. Next consider set inclusion. Let X be a subset of IR, then ∈ X transforms to ∗ ∈ ∗X. But ordinary set inclusion too is, of course, applicable to ∗X, so that there would be two set inclusions for the ∗transform ∗X of X. Fortunately, the two are identical, so that dropping the asterisk is a must. g) The logical connectives, and both quantiﬁers. For all of them the ∗transform is identical to the inverse image, so that asterisks should be dropped. h) Deﬁnitions. For example continuity transforms to ∗-continuity and ∗f, introduced in Section 1.4, is ∗-continuous at c if (1.1) is true. i) Statements. To some extent this covers, of course, case h). To ﬁnd the ∗transform of a statement (1.1), it should be formulated in such a way that each bound variable x occurs in some set inclusion of the form x ∈ X, not x ⊂ Y . Then the ∗-transform is obtained by replacing each constant and each free variable by its ∗-transform. As an example consider (1.1) and (1.1) deﬁned in Section 1.4. Note that if in (1.1) ’∈ IR’ would have been left out, something diﬀerent would have been obtained. This is one of the reasons why at the end of Section 1.3 it was suggested to explicitly include each bound variable in some set inclusion. Why the inclusion x ⊂ Y should be avoided will become clear in the next section.

One of the basic principles of nonstandard analysis is that any given classical statement (1.1) is true if and only if its ∗-transform is true, which results from (1.1) by replacing all its constants and free variables by their ∗-transforms. Note that the bound variables are not replaced. The principle is applied both ways, from IR to ∗IR, or from ∗IR to IR. In either case one says that the deduction is done by ‘transfer’. Assuming that everything is in prenex normal form, two simple nontrivial cases are,

∀x ∈ X : P(x,s) and ∃x ∈ X : P(x,s),

where X is some set and P(x,s) is some atomic substatement with x a free variable and s a constant or a free variable. The ∗-transforms are, ∀x ∈∗X : P(x,∗s) and ∃x ∈∗X : P(x,∗s), respectively. Clearly, for each of the two classical statements transfer is trivial in one direction, assuming that X is a subset of ∗X, but not necessarily in the opposite direction. The following two implications are the nontrivial ones, [∀x ∈ X : P(x,s)] ⇒ [∀x ∈∗X : P(x,∗s)], [∃x ∈∗X : P(x,∗x)] ⇒ [∃x ∈ X : P(x,s)]. Note that the ﬁrst implication starts from a classical statement and leads to its ∗-transform, whereas the second one starts from a ∗-transform and leads to the corresponding classical statement.

In by far the most practical situations applying transfer is fairly obvious. In what follows transfer is applied in a slightly complicated situation, where it is required to show the equivalence of statements (1.1) and (1.1), as well as that (1.1) can be simpliﬁed to Q, with (1.1), (1.1) and (1.1) as in Section 1.4. Trivially, by transfer, (1.1) and (1.1) are equivalent, so it remains to show the equivalence of (1.1) and (1.1). a) Let (1.1) be true, and let ε ∈ IR, ε < 0, and δ ∈ ∗IR, δ ' 0 be arbitrary. Then for some δ0 ∈ IR, δ0 > 0, ∀x ∈ IR, | x−c |< δ0 :| f(x)−f(c) |< ε, hence, by transfer, and because ∗c = c, ∗ε = ε, ∗δ0 = δ0, ∗f(c) = f(c), ∀x ∈∗IR,| x−c |< δ0 :|∗f(x)−∗f(c) |< ε. Let x = c+δ, then, because by deﬁnition of inﬁnitesimal | δ |< δ0, |∗f(c+ δ)−∗f(c) |< ε. But since ε is arbitrary, this means that ∗f(c+δ)−∗f(c) ' 0, and since δ is arbitrary that (1.1) is true. b) Conversely, let (1.1) be true, and let ε ∈ IR, ε > 0, and δ ∈ ∗IR, δ ∼ 0, δ > 0, be arbitrary. For each x ∈∗IR, | x−c |< δ it follows that x−c = δ0 for some δ0 ' 0, hence, by (1.1), ∗f(x)−∗f(c) ' 0, and, by deﬁnition of inﬁnitesimal, |∗f(x)−∗f(c) |< ε. Apparently, ∃δ00 ∈∗IR, δ00 > 0 : ∀x ∈∗IR,| x−c |< δ00 :|∗f(x)−∗f(c) |< ε, (take, for example, δ00 = δ), hence, by transfer (in the opposite direction as under a)), ∃δ00 ∈ IR, δ00 > 0 : ∀x ∈ IR,| x−c |< δ00 :| f(x)−f(c) |< ε. Since is arbitrary this proves (1.1).

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