珍贵数学文献（II）

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 deﬁnitions and proofs of elementary calculus can be simpliﬁed 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 simpliﬁed 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 deﬁned as a+jδ is limited. Then β is a (hence the) least upper bound of S. For ﬁrst 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
β0 ≥ s0 > a+jδ−δ = β+ε−δ > β0+1/m+ε−δ, or δ−ε > 1/m, a contradiction.
Note that this proof is not much shorter than its classical counterpart, that essentially runs as follows. Let a1 = a, b1 = b and n = 1. (∗) If S ⊆ [an, (an + bn)/2], then let an+1 = an, bn+1 = (an +bn)/2, otherwise let an+1 = (an +bn)/2, bn +1 = bn. In either case replace n by n+1 and start again from (∗). This procedure deﬁnes two concurrent Cauchy sequences (an) and (bn), both converging to the least upper bound of S (the reader may work out the details).
Since the theorem is wrong if IR is replaced byQ, both proofs must use something that is typical for IR. Indeed each limited element of ∗IR (not ∗Q) has a standard part, and any Cauchy sequence converges to some element of IR (not Q). This illustrates the obvious fact that a nonstandard proof must contain all essential steps – perhaps in disguise – of the corresponding classical proof.
3.2 Simplifying deﬁnitions and proofs of elementary calculus
First of all recall from Sections 1.4 and 1.5 that a function f from IR to IR is continuous at c ∈ IR if, ∀ε ∈ IR,ε > 0 : ∃δ ∈ IR,δ > 0 : ∀x ∈ IR,| x−c |< δ :| f(x)−f(c) |< ε or, equivalently, if, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,| x−c |< δ :|∗f(x)−∗f(c) |< ε or, equivalently, if, ∀δ ∈∗IR,δ ' 0 : ∗f(c + δ)−∗f(c) ' 0. The ﬁrst simpliﬁcation, therefore, reads as follows.
Theorem 3.2.1 (Simpliﬁed deﬁnition of the continuity of real-valued real functions.) f : IR → IR is continuous at c ∈ IR if and only if, ∀δ ∈∗IR,δ ' 0 : ∗f(c + δ)−∗f(c) ' 0.
103
Theorem 3.2.2 (Simpliﬁed deﬁnition of uniform continuity.) f : S → IR, S ⊆ IR is uniformly continuous in S if and only if, ∀x,y ∈∗S,x−y ' 0 : ∗f(x)−∗f(y) ' 0.
Proof: Recall that f : S → IR, S ⊆ IR is uniformly continuous in S if, ∀ε ∈ IR,ε > 0,∃δ ∈ IR,δ > 0 : ∀x,y ∈ S,| x−y |< δ :| f(x)−f(y) |< ε. By transfer, this is equivalent to, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0, ∀x,y ∈∗S,| x−y |< δ :|∗f(x)−∗f(y) |< ε. But this can be simpliﬁed to, ∀x,y ∈∗S,x−y ' 0 : ∗f(x)−∗f(y) ' 0. For let (3.1) be true and let m ∈ IN be given arbitrarily. Then there exist ε ∈ IR, ε > 0 such that ε < 1/m, and δ as in (3.1). Hence ∀x,y ∈ S, | x − y |< δ : | f(x)−f(y) |< 1/m, or, by transfer, ∀x,y ∈∗S,| x−y |< δ :|∗f(x)−∗f(y) |< 1/m, so that, as m was arbitrary, ∗f(x) − ∗f(y) ' 0, which proves (3.1), since in absolute value any inﬁnitesimal is smaller than δ. Conversely, let (3.1) be true, let ε ∈ IR, ε > 0 be arbitrary and let δ ' 0, δ > 0. Hence if x,y ∈∗S, | x−y |< δ then ∗f(x)−∗f(y) = ε0 for some ε0 ' 0. In other words, since | ε0 |< ε, ∃δ0 ∈∗IR,δ0 > 0 : ∀x,y ∈∗S,| x−y |< δ0 :|∗f(x)−∗f(y) |< ε, (take, for example δ0 = δ) or, by transfer (in the opposite direction), ∃δ0 ∈ IR,δ0 > 0 : ∀x,y ∈ S,| x−y |< δ0 :|∗f(x)−∗f(y) |< ε, which proves (3.1) since ε was arbitrary.
Theorem 3.2.3 If f is continuous at each x ∈ [a,b], a, b ∈ IR, a < b, then f is uniformly continuous in [a,b].
104
Simpliﬁed proof: Let x,y be in ∗[a,b], then by Theorem 2.13.3, st(x) ∈ [a,b] and st(y) ∈ [a,b]. Let x−y ' 0. Since y−st(y) ' 0 it follows that x−st(y) ' 0. By continuity, ∗f(y)−∗f(st(y)) ' 0 and ∗f(x)−∗f(st(y)) ' 0, so that ∗f(x)−∗f(y) ' 0.
Theorem 3.2.4 (Simpliﬁed limit deﬁnition.) Let f : IR → IR, then lim x→c f(x) = k, c,k ∈ IR if and only if, ∀δ ∈∗IR,δ ∼ 0 : ∗f(c + δ)−k ' 0, so that k =st[∗f(c + δ)].
Proof: By deﬁnition, the limit exists if and only if, ∀ε ∈ IR,ε > 0 : ∃δ ∈ IR,δ > 0 : ∀x ∈ IR,0 <| x−c |< δ :| f(x)−k |< ε, or, by transfer, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,0 <| x−c |< δ :|∗f(x)−k |< ε, which can be simpliﬁed to, ∀δ ∈∗IR,δ ∼ 0 : ∗f(c + δ)−k ' 0.
Exercise: Complete this proof. Now let f : IN → IR, so that f is an inﬁnite sequence, and let c be replaced by ∞.
Theorem 3.2.5 (Another simpliﬁed limit deﬁnition.) Let f : IN → IR and k ∈ IR, then lim x→∞ f(n) = k, k ∈ IR, if and only if, ∀n ∈∗IN, n ∼∞ : ∗f(n)−k ' 0, so that k =st[∗f(n)].
Proof: By deﬁnition, the limit exists if and only if, ∀ε ∈ IR,ε > 0 : ∃n0 ∈ IN : ∀n ∈ IN,n > n0 :| f(n)−k |< ε, or, by transfer, ∀ε ∈∗IR,ε > 0 : ∃n0 ∈∗ N : ∀n ∈∗IN,n > n0 :|∗f(n)−k |< ε,
105
which can be simpliﬁed to, ∀n ∈∗IN,n ∼∞ : ∗f(n)−k ' 0.
Exercise: Again complete the proof.
Exercise: Treat the cases where the limit itself is inﬁnite.
Theorem 3.2.6 If f is a nondecreasing inﬁnite sequence, that is bounded above, then f(n) has a ﬁnite limit for n tending to ∞.
Classical proof: The set {f(n) : n ∈ IN} is bounded above, hence in IR has a least upper bound β, so that f(n) ≤ β for all n ∈ IN, and for each m ∈ IN, f(n0) > β−1/m for some n0 ∈ IN, and since f is nondecreasing this implies that lim n→∞ f(n) = β. Exercise: Give a direct nonstandard proof similar to that of Theorem 3.1.1 (the least upper bound theorem), not using Theorem 3.1.1.
Theorem 3.2.7 (The intermediate value theorem.) If a, b ∈ IR, a < b, and f(a) < 0, f(b) > 0, then f(c) = 0 for some c, a < c < b.
Simpliﬁed proof: See Section 1.4.
Theorem 3.2.8 (The extreme value theorem.) Let f : [a,b] → IR, a,b ∈ IR, a < b, and let f be continuous at each point of [a,b]. Then f(x) ≤ f(c) for some c ∈ [a,b] and all x ∈ [a,b], i.e. f has a maximum somewhere in the closed interval between a and b. And similarly for minimum.
Simpliﬁed proof: Let ω ∈ ∗IN, ω ∼∞, be arbitrary, and divide ∗[a,b] in ω equal subintervals of length δ = (b − a)/ω. Let n ∈ ∗IN be such that ∗f(a + nδ) ≥ ∗f(a+iδ) for all i = 0,1,...,ω. The existence of n follows by transfer, since any ﬁnite set has a maximum, hence so has any hyperﬁnite set. Obviously, a + nδ is limited, hence c =st(a+nδ) is well deﬁned and by continuity, ∗f(a+nδ)−f(c) = ε for some ε ' 0. Each x ∈ [a,b] is within the distance δ of some a + iδ and δ ∼ 0, hence, again by continuity, f(x) = ∗f(a + iδ) + ε0 for some ε0 ' 0, hence, f(x) ≤∗f(a + nδ) + ε0 = f(c) + ε + ε0, i.e. f(x) ≤ f(c).
106
Theorem 3.2.9 (The composite function theorem.) Let g(w) be deﬁned for w in a neighborhood of c ∈ IR, and let f(x) be deﬁned for x in a neighborhood of g(c). Then f◦g is continuous at c if g is continuous at c and f is continuous at g(c).
Simpliﬁed proof: Let δ ' 0, then ∗g(c + δ) − g(c) ' 0, hence it follows that ∗f(∗g(c + δ))−f(g(c)) ' 0.
3.3 Continuity and limits for internal functions
So far nonstandard characterizations were given for continuity and limits of classical functions. How about arbitrary internal functions? Let f be some internal function from ∗IR to ∗IR, and let c ∈∗IR, so that c may be hyperlarge or ‘almost standard’ i.e. be the sum of a real number and a nonzero inﬁnitesimal. Or rather, let F be the set of all classical functions from IR to IR and let f ∈ ∗F. Then, by deﬁnition D) of Section 2.9, for suitable fi and ci, f = H(fi) is ∗continuous at c = H(ci) if for all i ∈ IN, fi is continuous at ci. Here fi : IR → IR and ci ∈ IR.
Theorem 3.3.1 (Continuity of internal functions.) The internal f : ∗IR →∗IR is ∗continuous at c ∈∗IR if and only if in the classical deﬁnition IR is replaced by ∗IR, i.e. if, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,| x−c |< δ :| f(x)−f(c) |< ε.
Proof: Letting f = H(fi) and c = H(ci), f is ∗continuous at c if and only if, H[∀εi ∈ IR,εi > 0 : ∃δi ∈ IR,δi > 0 : ∀xi ∈ IR,| xi −ci |< δi : | fi(xi)−fi(ci) |< εi]. By L oˇs’ theorem (Theorem 2.7.1) this is equivalent to, ∀H(εi) ∈∗IR,H(εi) > 0 : ∃H(δi) ∈∗IR,H(δi) > 0 : ∀H(xi) ∈∗IR,| H(xi)−H(ci) |< H(δi) : | H(fi)(H(xi))−H(fi)(H(ci)) |< H(εi) and hence to what has to be proved. Warning: If f or c is nonstandard, ∗continuity is not always equivalent to, ∀δ ∈∗IR,δ ' 0 : f(c + δ)−f(c) ' 0.
Counterexamples:
107
a) c standard, but f nonstandard; let ω ∼∞, f(x) = ωx, c = 1, δ = 1/ω1/2; then f(c + δ)−f(c) = ωδ ∼∞. b) f standard, but c nonstandard; let f(x) = x2, δ ∼ 0, c = 1/δ; then f(c + δ)−f(c) = 2 + δ2 ∼ 2. Yet, ∀δ ∈ ∗IR, δ ' 0 : f(c + δ)−f(c) ' 0 makes sense for arbitrary internal f and c. If this is true, then f is called S-continuous at c.
Examples: a) Let α be a positive inﬁnitesimal, and f(x) = αx if x ≥ 0, f(x) = 0 if x < 0. Then f is S-continuous everywhere in ∗IR. It is ∗continuous at c ∈∗IR if c 6= 0, but not at c = 0. b) Let ω ∼∞ and f(x) = ωx. Then f is nowhere S-continuous, since f(x)− f(c) = ω(x−c) = ω1/2 if x−c = ω−1/2 ∼ 0. It is continuous everywhere in ∗IR. c) f(x) = x2. Then f is not S-continuous if c ∼∞. It is ∗continuous everywhere in ∗IR. Theorem 3.3.2 The internal function f : ∗IR → ∗IR is S-continuous at c ∈ ∗IR if and only if, ∀ε ∈ IR,ε < 0 : ∃δ ∈ IR,δ > 0 : ∀x ∈∗IR,| x−c |< δ :| f(x)−f(c) |< ε. (Note that both ε and δ are standard, but that x is internal.)
Proof: The if part. Let ε ∈ IR, ε > 0 and δ ∈∗IR, δ ' 0 be given arbitrarily. Then there is a δ0 ∈ IR, δ0 > 0 such that, ∀x ∈∗IR,| x−c |< δ0 :| f(x)−f(c) |< ε. As | δ |< δ0, so that | x−c |< δ0 if x = c+δ, it follows that | f(c+δ)−f(c) |< ε, and since ε is arbitrary that f(c + δ)−f(c) ' 0. The only-if part. Conversely, let ε and δ be as before but such that δ > 0. Then ∀x ∈∗IR, | x−c |< δ : | f(x)−f(c) |< ε. Now let the set S be deﬁned by, S = {δ ∈∗IR : δ > 0 and ∀x ∈∗IR,| x−c |< δ :| f(x)−f(c) |< ε}. Then if δ ∈ S every number between 0 and δ is in S. By the internal deﬁnition principle (Corollary 2.7.1) S is internal, but it contains as a subset the set of all positive inﬁnitesimals and the latter is external (see Theorems 2.10.3 and 2.11.4),
108
so that S must contain some δ > 0 that is not an inﬁnitesimal, and it follows that S must contain some δ > 0, δ ∈ IR. Remark: In this proof the fact that an external set is not internal has been used. This fact is called Cauchy’s principle. It is an example of a principle of permanence. In general this is a statement that if some set S contains some subset T, the latter is strictly contained in S, because S and T happen to be diﬀerent kinds of set. In classical mathematics such principles do not seem to play a real part, but in nonstandard mathematics there are several of them, although there are only a few primary forms, or perhaps only one, i.e. Cauchy’s principle, which obviously really is a matter of deﬁnition. See Section 4.1 for more details. In a similar way the ∗-transform of uniform continuity can be introduced: simply copy the classical deﬁnition and replace IR by ∗IR. And the simpliﬁed form of uniform continuity leads to S-uniform continuity. Hence the internal f : ∗IR →∗IR is S-uniformly continuous in ∗IR if, ∀x,y ∈∗IR,x−y ' 0 : f(x)−f(y) ' 0.
Exercise: Formulate and show a theorem similar to Theorem 3.3.2, but for uniform continuity. It is generally agreed to drop the asterisks in both ∗continuity and ∗uniform continuity, as well as in similar indications. But keep in mind that S-continuity is then not a special form of continuity, and similarly for other attributes. Turning to ∗limits, the deﬁnitions similar to those in Section 3.2 are, dropping the asterisks: the internal f : ∗IR → ∗IR tends to the limit k ∈ ∗IR for x ∈ ∗IR tending to c ∈∗IR, if, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,0 <| x−c |< δ :| f(x)−k |< ε. And the internal f : ∗IN → ∗IR tends to the limit k ∈ ∗IR for n ∈ ∗IN tending to inﬁnity, if, ∀ε ∈∗IR,ε > 0 : ∃n0 ∈∗IN : ∀n ∈∗IN,n > n0 :| f(n)−k |< ε. Similar to S-continuity the deﬁnitions of S-limit are as follows. The internal f : ∗IR →∗IR tends to the S-limit k ∈∗IR for x ∈∗IR tending to c ∈∗IR, if, ∀δ ∼ 0 : f(c + δ)−k ' 0. And the internal f : ∗IN → ∗IR tends to the S-limit k ∈ ∗IR for n ∈ ∗IN tending to inﬁnity, if, ∀n0 ∈∗IN,n0 ∼∞ : f(n0)−k ' 0.
109
Theorem 3.3.3 The internal f : ∗IR → ∗IR tends to the S-limit k ∈ ∗IR for x ∈∗IR tending to c ∈∗IR, if and only if, ∀ε ∈ IR,ε > 0 : ∃δ ∈ IR,δ > 0 : ∀x ∈∗IR,0 <| x−c |< δ :| f(x)−k |< ε.
Proof: Left as an exercise. Theorem 3.3.4 The internal f : ∗IN → ∗IR tends to the S-limit k ∈ ∗IR for n ∈∗IN tending to inﬁnity, if and only if, ∀ε ∈ IR,ε > 0 : ∃n0 ∈ IN : ∀n ∈∗IN,n > n0 :| f(n)−k |< ε.
Proof: Left as an exercise.
A special case arises when k is ﬁnite, because then st(k) is well deﬁned. Then also f(x) or f(n) are ﬁnite for x close enough to c or n large enough. Assuming in the ﬁrst case that c is ﬁnite as well, it follows that the classical limits for x tending to st(c) or n tending to ∞ are equal to st(k).
Theorem 3.3.5 Let f, c and k be as before, and let k and c be ﬁnite. If f(x) tends to k for x ∈∗IR tending to c, then, lim x→st(c) st(f(x)) = st(k), where x ∈ IR, and if f(n) tends to k for n ∈∗IN tending to ∞, then, lim n→∞ st(f(n)) = st(k), where n ∈ IN.
Proof: In the ﬁrst case
∃δ1 ∈ IR,δ1 > 0 : ∀x ∈∗IR,0 <| x−c |< δ1 :| f(x)−k |< 1, so that ∀x ∈ IR, 0 <| x−st(c) |< δ1/2 :| f(x) |<|st(k) | +2, which means that f(x) is ﬁnite for these x’s, from which the ﬁrst claim follows. The second claim is shown in a similar way.
Exercise: Treat the cases where the limit itself is inﬁnite. And deﬁne the corresponding S-limits.
110
3.4 More nonstandard characterizations of classical notions
Theorem 3.4.1 (Nonstandard characterization of Cauchy sequence.) s(n) is a classical Cauchy sequence if and only if, ∀n,p ∈∗IN,n,p ∼∞ : ∗s(n)−∗s(p) ' 0.
Proof: By deﬁnition, ∀m ∈ IN : ∃k ∈ IN : ∀n,p ∈ IN,n,p > k :| s(n)−s(p) |< 1/m, and by transfer, ﬁxing m ∈ IN and k ∈ IN, ∀n,p ∈ IN,n,p > k :| s(n)−s(p) |< 1/m, is equivalent to, ∀n,p ∈∗IN,n,p > k :|∗s(n)−∗s(p) |< 1/m. Now let n,p ∼∞, so that automatically n,p > k, no matter the value of k ∈∗IN, then (3.1) implies that, ∀m ∈ IN : ∃k ∈ IN : n,p ∈∗IN,n,p ∼∞ :|∗s(n)−∗s(p) |< 1/m. But since k plays no part any more this can be simpliﬁed to, ∀m ∈ IN : ∀n,p ∈∗IN,n,p ∼∞ :|∗s(n)−∗s(p) |< 1/m, hence to, ∀n,p ∈∗IN,n,p ∼∞ : ∀m ∈ IN :|∗s(n)−∗s(p) |< 1/m, hence to, ∀n,p ∈∗IN,n,p ∼∞ : ∗s(n)−∗s(p) ' 0, which is (3.1).
Conversely, consider the negation of (3.5), that is, ∃m ∈ IN : ∀k ∈ IN : ∃n,p ∈ IN,n,p > k :| s(n)−s(p) |≥ 1/m,
111
ﬁx m ∈ IN and apply transfer to, ∀k ∈ IN : ∃n,p ∈ IN,n,p > k :| s(n)−s(p) |≥ 1/m, giving, ∀k ∈∗IN : ∃n,p ∈∗IN,n,p > k :|∗s(n)−∗s(p) |≥ 1/m, which implies, ﬁxing k ∼∞ arbitrarily, that n,p ∼∞, hence, ∃n,p ∈∗IN,n,p ∼∞ :|∗s(n)−∗s(p) |≥ 1/m, so that (3.1) implies that, ∃m ∈ IN : ∃n,p ∈∗IN,n,p ∼∞ :|∗s(n)−∗s(p) |≥ 1/m, or, ∃n,p ∈∗IN,n,p ∼∞ : ∃m ∈ IN :|∗s(n)−∗s(p) |≥ 1/m, or, ∃n,p ∈∗IN,n,p ∼∞ : ¬[∗s(n)−∗s(p) ' 0], which is the negation of (3.4).
Theorem 3.4.2 (Nonstandard characterization of bounded set.) Let L be the set of all limited elements of ∗IR. Then S ⊆ IR is bounded if and only if ∗S ⊆ L.
Proof: S is bounded if,
∃m ∈ IN : ∀s ∈ S :| s |≤ m,
hence, by transfer, if,
∃m ∈ IN : ∀s ∈∗S :| s |≤ m,
so that ∗S ⊆ L. Conversely, if S is not bounded then, ∀m ∈ IN : ∃s ∈ S :| s |> m, hence, by transfer, ∀m ∈∗IN : ∃s ∈∗S :| s |> m, and taking m hyperlarge it follows that | s | is hyperlarge for some s ∈ ∗S, so that s 6∈ L.
112
Theorem 3.4.3 (Nonstandard characterization of open set.) Let S ⊆ IR and let h(S) = {t ∈ ∗IR : t ' s for some s ∈ S}. Then S is open if and only if, h(S) ⊆∗S.
Proof: S is open if, ∀s ∈ S : ∃m ∈ IN : ∀t ∈ IR,| t−s |< 1/m : t ∈ S, hence, by transfer, ∀s ∈ S : ∃m ∈ IN : ∀t ∈∗IR,| t−s |< 1/m : t ∈∗S, so that, restricting t such that t ' s, ∀s ∈ S : ∀t ∈∗IR,t ' s : t ∈∗S, i.e. h(S) ⊆∗S.
Conversely, if S is not open, then, ∃s ∈ S : ∀m ∈ IN : ∃t ∈ IR,| t−s |< 1/m : t 6∈ S, hence, by transfer, ∃s ∈ S : ∀m ∈∗IN : ∃t ∈∗IR,| t−s |< 1/m : t 6∈∗S, and taking m hyperlarge it follows that for some s ∈ S and some t ∈∗IR we have that t ' s, but t 6∈∗S, hence that h(S) is not a subset of ∗S. Remark: h(S) is called the halo (or the monad) of S.
Theorem 3.4.4 (Nonstandard characterization of closed set.) S ⊆ IR is closed if and only if, h(Sc) ⊆∗(Sc) = (∗S)c.
Proof: Follows directly from the previous theorem.
Exercise: Show the last theorem independently of the previous theorem.
Theorem 3.4.5 (Nonstandard characterization of interior point.) Let s ∈ IR and let h(s) = {t ∈ IR : t ' s}. Then s is an interior point of S ⊆ IR if and only if h(s) ⊆∗S.
113
Proof: Since s is an interior point of S if, ∃m ∈ IN : ∀t ∈ IR,| t−s |< 1/m : t ∈ S, the proof is a simpliﬁed version of that of Theorem 3.4.3. The details are left as an exercise.
Remark: In view of the previous remark, h(s) is of course called the halo of the point s. Note that h(0) is the set of all inﬁnitesimals.
Theorem 3.4.6 (Nonstandard characterization of boundary point.) s ∈ IR is a boundary point of S ⊆ IR if and only if both h(s)∩∗S and h(s)∩(∗S)c are nonempty.
Proof: If s is a boundary point of S then, ∀m ∈ IN : [∃t ∈ IR,| t−s |< 1/m : t ∈ S]∧[∃t ∈ IR,| t−s |< 1/m : t 6∈ S] hence, by transfer, ∀m ∈∗IN : [∃t ∈∗IR,| t−s |< 1/m : t ∈∗S]∧[∃t ∈∗IR,| t−s |< 1/m : t 6∈∗S] so that, taking m ∼∞, [∃t : t ' s,t ∈∗S]∧[∃t : t ' s,t 6∈∗S], hence h(s)∩∗S 6= ∅ and h(s)∩(∗S)c 6= ∅. Conversely, if s is not a boundary point of S, then, ∃m ∈ IN : [∀t ∈ IR,| t−s |< 1/m : t 6∈ S]∨[∀t ∈ IR,| t−s |< 1/m : t ∈ S], hence, by transfer, ∃m ∈ IN : [∀t ∈∗IR,| t−s |< 1/m : t 6∈∗S]∨[∀t ∈∗IR,| t−s |< 1/m : t ∈∗S]. The ﬁrst substatement between square brackets implies that if t ' s then t 6∈∗S, so that h(s) ⊆ (∗S)c, and the second one similarly that h(s) ⊆ ∗S, so that either h(s)∩∗S = ∅ or h(s)∩(∗S)c = ∅.
Theorem 3.4.7 (Nonstandard characterizations of accumulation point and closure.) s ∈ IR is an accumulation point (or limit point) of S ⊆ IR if and only if, ∃t ∈∗S,t 6= s : t ∼ s. Let cl S be the closure of S. Then s ∈ cl S if and only if, ∃t ∈∗S : t ' s.
114
Proof: If s is an accumulation point of S then, ∀m ∈ IN : ∃t ∈ S,t 6= s :| t−s |< 1/m, hence, by transfer, ∀m ∈∗IN : ∃t ∈∗S,t 6= s :| t−s |< 1/m, so that, taking m hyperlarge, ∃t ∈∗S, t 6= s: t ∼ s. Conversely, if s is not an accumulation point of S, then, ∃m ∈ IN : ∀t ∈ S,t 6= s :| t−s |≥ 1/m, hence, by transfer, ∃m ∈ IN : ∀t ∈∗S,t 6= s :| t−s |≥ 1/m, so that, ∀t ∈∗S,t 6= s : ¬[t ' s]. The second part of the theorem follows by observing that s ∈ cl S if and only if s ∈ S or else if s is an accumulation point of S. Exercise: Give an alternative proof of Theorem 3.4.4, using the fact that S is closed if and only if S = cl S.
3.5 Inverse functions; bc
Recall that a function f : S → T has an inverse f−1 if and only if f is bijective, and then f−1(t) = s if f(s) = t.
Theorem 3.5.1 Let a function f be monotonically increasing (or decreasing) and be continuous in [a,b], a,b ∈ IR, a < b. Then,
1) range (f), the range of f, is a ﬁnite closed interval, 1) f has an inverse, 1) f−1 too is monotonically increasing (or decreasing), and 1) f−1 is continuous in its domain.
Proof: Only the case where f is increasing is considered.
115
1) As a ≤ x ≤ b implies that f(a) ≤ f(x) ≤ f(b), range (f) ⊆ [f(a),f(b)]. And if f(a) ≤ w ≤ f(b) then by the intermediate value theorem (Theorem 3.2.7) there is a c ∈ [a,b] such that f(c) = w which means that [f(a),f(b)] ⊆ range (f). Therefore, range (f) = [f(a),f(b)]. 2, 3) Clearly, f−1 exists, [f(a),f(b)] is its domain, and it is increasing. 4) Let w = f(c) ∈ [f(a),f(b)], and let ε ∼ 0. If ε > 0 then, of course, w must be smaller than f(b), and if ε < 0 then w must be larger than f(a). Assume ε > 0. Note that ∗(f−1) = (∗f)−1, so that parentheses are not required here. Let c0 = ∗f−1(w + ε) so that c0 > c. If c0 were not inﬁnitesimally close to c, then c0 > c+1/m for some m ∈ IN. As c,m ∈ IR. f(c + 1/m) > f(c) + 1/n for some n ∈ IN, so that ∗f(c0) > f(c + 1/m) > f(c)+1/n = w+1/n, but ∗f(c0) = w+ε, hence ε > 1/n, a contradiction. It follows that c0 ∼ c, which shows the continuity of f−1 at w. The next subject of this section is the introduction of bc, for b,c ∈ IR, b > 0. If c ∈Q this can best be done in the classical way, using the functions xc and bx as well the properties of inverse functions. Hence begin with xn, n ∈ IN, x ∈ IR, x > 0, which is monotonically increasing and continuous, leading to the deﬁnition of x1/n as its inverse, and hence to xm/n, m,n ∈ IN, x ∈ IR, x > 0, either as (x1/n)m or as (xm)1/n. To see that the two are identical, note that (yn)1/n = y, so that, taking y = (x1/n)m it follows that,
(((x1/n)m)n)1/n = (x1/n)m,
and taking y = x it follows that,
(((x1/n)m)n)1/n = ((x)m)1/n = (xm)1/n. For c ∈Q, c > 0, let xc = 1/x−c, and in view of xc •xd = xc+d, let x0 = 1. Then xc, c ∈Q, x ∈ IR, x > 0 is increasing if c > 0, decreasing if c > 0, and equal to 1 if c = 0. Next consider bx, b ∈ IR, b > 0, x ∈Q, which is now well deﬁned, then bx is increasing if b > 1, decreasing if b < 1, and equal to 1 if b = 1; and continuous at each x ∈Q. For c ∈ IR there is a nonstandard alternative. Given any c ∈ IR, by Theorem 2.13.2, c = st(c0) for some c0 ∈ ∗Q, where c0 is determined uniquely up to a hyperrational inﬁnitesimal. Now let,
g(c) = st(bc0),
then g(c) = bc, where, bc = lim x→c
bx, x ∈Q.
116
To see this note ﬁrst of all that g(c) is deﬁned uniquely, for if ε ' 0, ε ∈∗Q, then st(bc0+ε) = st(bc0)• st(bε) = st(bc0)•1. Now let d ∈Q, such that c = d+r for some r > 0, then g(d) = bd. Since c0 = c+ε for some ε ' 0, c0 = d + r + ε and, g(c)−g(d) = st(bd+r+ε)−bd = bd •[st(br+ε)−1]. In this product the ﬁrst factor is positive, and the second one is positive (or negative) if b > 1 (or b < 1), and zero if b = 1. A similar result is obtained if r < 0. Hence the function g is monotonous if not constant, so that if c is between the rationals d and e, then g(c) is between bd and be. This shows both the monotonicity and the continuity of g(x), x ∈ IR, and hence that g(c) = bc.
Exercise: Show that st(p•q) = st(p)• st(q) and that st(bε) = 1 if b > 0 and ε ' 0, ε ∈∗Q.
3.6 Diﬀerentiation
Let f : ∗IR → ∗IR, c ∈ ∗IR, then f is said to be diﬀerentiable at c if for some k ∈∗IR, lim x→c f(x)−f(c) x−c exists and is equal to k. Then this limit, which obviously is a ∗limit, is called the derivative of f at c, and usually k is replaced by f0(c) or by df(c) dx . In case everything is standard, this deﬁnition becomes the classical deﬁnition of diﬀerentiability, and from Section 3.3 it follows that f : IR → IR is diﬀerentiable at c ∈ IR if for some k ∈ IR, ∀δ ∼ 0 : ∗f(c + δ)−f(c) δ ' k = f0(c), so that, f0(c) = st"∗f(c + δ)−f(c) δ #. This means that f0(c) is inﬁnitesimally close to a quotient, justifying to a certain extent calling f0(c) a diﬀerential quotient, even though f0(c) is a limit of a quotient.
117
As before this need not be true if f, and c are internal; then f is called Sdiﬀerentiable at c if for some internal k,
∀δ ∼ 0 :
f(c + δ)−f(c) δ ' k = f0(c). From the deﬁnition of f0(c) in the standard case it follows that, ∀δ ' 0 : ∗f(c + δ)−f(c)−f0(c)•δ = τδ, for some τ ' 0. This is known as the increment theorem. Conversely, if, for some k ∈ IR and, ∀δ ' 0 : ∗f(c + δ)−f(c)−kδ = τδ, for some τ ' 0, then f : IR → IR is diﬀerentiable at c ∈ IR and f0(c) = k.
Theorem 3.6.1
1) If f : IR → IR is diﬀerentiable at c ∈ IR, then f is continuous at c. 1) If f is diﬀerentiable at c, and b ∈ IR, then f +b and b•f are diﬀerentiable at c as well, and, (f + b)0(c) = f0(c),(b•f)0(c) = b•f0(c). 1) If both f and g are diﬀerentiable at c, then (f + g)0(c) = f0(c) + g0(c),(f •g)0(c) = f0(c)•g(c) + f(c)•g0(c), and, if g(c) 6= 0, then, (f/g)0(c) = (f0(c)•g(c)−f(c)•g0(c))/g2(c). Note that in the last case that g(x) 6= 0 if x = c + δ, for δ ' 0, as g is continuous at c. In particular, if g(x) = 1/f(x), and f(c) 6= 0, then, g0(c) = −f0(c)/f2(c). 1) Chain rule. If g is diﬀerentiable at c, and f is diﬀerentiable at g(c), then the composite function F = f◦g, too is diﬀerentiable at c, and, F0(c) = (f◦g)0(c) = f0(g(c))•g0(c). Proof: 1) Follows immediately from ∗f(c + δ)−f(c)−f0(c)•δ = τδ, so that, ∗f(c + δ)−f(c) ' 0 if δ ' 0.
118
2) and 3) are left as exercises. 4) For any δ ' 0, and for some µ ' 0, ∗f(g(c) + δ)−f(g(c))−f0(g(c))•δ = µδ Let, given any ε ' 0, δ = ∗g(c + ε)−g(c) = g0(c)•ε + τε, for some τ ' 0. Then δ ' 0 and, ∗f(∗g(c + ε))−f(g(c))−f0(g(c))•(g0(c) + τ)•ε = µδ, hence, ∗f(∗g(c + ε))−f(g(c))−f0(g(c))•g0(c)•ε = f0(g(c))•τε + µ(g0(c)•ε + τε) = τ0ε, for some τ0 ' 0, which means that f◦g is diﬀerentiable at c, and that its derivative at c is equal to f0(g(c))•g0(c).
Theorem 3.6.2 (The critical point theorem.) Let X be an interval of IR, c ∈ X, f : X → IR be continuous at each x ∈ X, and c be a maximum or a minimum of f over X. Then c is an endpoint of X, or f0(c) is undeﬁned, or f0(c) = 0.
Proof: Let c be a maximum of f over X. If c is not an endpoint of X and if f0(c) exists, then, ∀δ ∼ 0 : f0(c) = st"∗f(c + δ)−f(c) δ #. Taking ﬁrst δ positive and then negative this gives that f0(c) ≤ 0 and f0(c) ≥ 0, hence f0(c) = 0.
Theorem 3.6.3 (Rolle’s theorem.) Let f be continuous in [a,b], a,b ∈ IR, a < b, and be diﬀerentiable in (a,b). Moreover, let f(a) = f(b) = 0. Then f0(c) = 0 for some c ∈ (a,b).
Proof: The proof is classical in nature, and relies on Theorems 3.2.8 and 3.6.2.
The proofs of the next corollaries too are classical in nature, and therefore are not given in detail.
119
Corollary 3.6.1 (Mean value theorem.) Let f be as in the previous theorem, except f(a) = f(b) = 0 need not be true. Then, f0(c) = f(b)−f(a) b−a for some c ∈ (a,b).
Proof: The proof follows by applying Rolle’s theorem to g, deﬁned by,
g(x) = f(x)−f(a)•
f(b)−f(a) b−a •(x−a).
Corollary 3.6.2 (Generalized mean value theorem.) Let f and g both be continuous in [a,b], and be diﬀerentiable in (a,b). In addition let g0(x) 6= 0 for all x ∈ (a,b). Then, f0(c) g0(c) = f(b)−f(a) g(b)−g(a) for some c ∈ (a,b).
Proof: The proof follows by applying Rolle’s theorem to h, deﬁned by,
h(x) = f(x)(g(b)−g(a))−g(x)(f(b)−f(a)).
Corollary 3.6.3 (Taylor’s theorem.) Let f and f0 both be continuous in [a,b], and let f00 exist in (a,b), then, f(b) = f(a) + f0(a)(b−a)/1! + f00(c)(b−a)2/2!, for some c ∈ (a,b), and similarly for higher derivatives.
Proof: The proof is based on the previous corollary.
Theorem 3.6.4 (L’Hopital’s theorem). Let both f and g be diﬀerentiable, and g0(x) 6= 0 in a neighborhood (a,b) of
120
c ∈ IR with the possible exception of the point c itself. Also assume that lim x→c f(x) = lim x→c g(x) = 0, and that lim x→c f0(x) g0(x) exists and is equal to k. Then,
lim x→c
f(x) g(x)
= k.
Similar statements hold true if k and/or c are replaced by ∞.
Proof: Only the case as stated will be proved. Without loss of generality it may be assumed that f(x) and g(x) are deﬁned at x = c and that f(x) = g(x) = 0, so that f and g are continuous at c. It follows that the generalized mean value theorem can be applied to both [c,c + δ] and [c−δ,c], for δ > 0 and close enough to zero. Taking δ ∼ 0 this gives that, for some c1 and c2, c < c1 < c + δ, c−δ < c2 < c, f0(c1) g0(c1) = f(c + δ)−f(c) g(c + δ)−g(c) and f0(c2) g0(c2) = f(c)−f(c−δ) g(c)−g(c−δ) . Since both c1 and c2 are inﬁnitesimally close to c this implies that these two expressions are inﬁnitesimally close to k and the theorem follows.
3.7 Integration
Only Riemann integration of continuous functions will be considered, even though the extension to more general functions is not very diﬃcult. Let a,b ∈ ∗IR, a < b and let f : [a,b] → ∗IR be a continuous function. Then, by deﬁnition, the Riemann integral of f over [a,b] is, J =Zb a f(x)dx = lim n→∞ n X i=1 f(a + i(b−a)/n)(b−a)/n. In case everything is standard, then, given any ω ∈∗IN, ω ∼∞, J = st"ω X i=1 f(a + i(b−a)/ω)(b−a)/ω#. Note that in this case dx = (b−a)/ω ∼ 0 and dx > 0. And the usual formulation is, J = st"ω X i=1 f(a + i•dx)/dx#.
121
Instead of the point a + i•dx any other point xi ∈ [a + (i−1)dx,a + i•dx] may be selected, i = 1,...,ω and J becomes st(S), where,
S =
ω X i=1
f(xi)dx,
a so-called Riemann sum. That S is ﬁnite follows from the extreme value theorem (Theorem 3.2.8), as this theorem implies the existence of m,M ∈ IR such that, ∀x ∈ [a,b] : m ≤ f(x) ≤ M, so that m(b−a) ≤ S ≤ M(b−a) and S is indeed ﬁnite. The notation of the integral correctly suggests that J depends neither on ω, nor on how the xi are selected. Indeed, let yi some other element of the interval [a + (i−1)dx,a + i•dx], then if ω is not changed it follows from the uniform continuity of f and dx < δ for all δ ∈ IR, δ > 0, that, ∀ε ∈ IR,ε > 0 :

ω X i=1 f(xi)dx− ω X i=1 f(yi)dx

< ω•ε•dx = ε(b−a), hence that the expression to the left of the last inequality is an inﬁnitesimal, which shows that J does not depend on the selection of the xi. And if ω1,ω2 ∼∞, and for k = 1,2, dxk = (b−a)/ωk and,
Sk =
ωk X i=1
f(a + i•dxk)dxk,
let ω = ω1ω2 and let dx and S be as before, then,
S −S1 =
ω1 X i=1
ω2 X j=1
[f(a +{(i−1)ω2 + j}dx)dx−f(a + iω2dx)dx], and again this is an inﬁnitesimal, as is S −S2, and hence so is S1 −S2. In case a > b, then by deﬁnitionRb a f(x)dx = −Ra b f(x)dx, and if a = b, then Ra a f(x)dx = 0. As remarked before f need not be continuous in order for J to exist. Also in what follows derivatives need not always be continuous, but let us not go into the details as this chapter is only concerned with showing the possibilities of nonstandard analysis.
The deﬁnition of improper integrals is as in classical analysis: just consider the appropriate limits.
122
Exercise: Present the details of introducing improper integrals, complete with the simpliﬁed forms in case everything is standard.
In the remainder of this section it is assumed that everything is standard.
Theorem 3.7.1 If a < b < c, then, Zc a f(x)dx =Zb a
f(x)dx +Zc b
f(x)dx.
Proof: Follows from the fact that the standard part of a sum is equal to the sum of the standard parts. Theorem 3.7.2 F deﬁned by F(x) = Rx a f(t)dt, a ≤ x ≤ b, a < b, is contin-uous. Moreover, the derivative of F(x) exists and is equal to f(x), a < x < b. Conversely, if G is such that G0(x) = f(x), then, F(b) =Zb a f(x)dx = G(b)−G(a).
Proof: The continuity of F follows from the fact that, if δ ' 0, Zx+δ a f(t)dt−Zx a f(t)dt =Zx+δ x f(t)dt, no matter whether δ is nonnegative or negative, in which case the right-hand side too is an inﬁnitesimal.
The second part of the theorem follows from the fact that m and M exist as before such that, m•dx ≤ F(x + dx)−F(x) =Zx+dx x f(t)dt ≤ M •dx. If dx ∼ 0, dx > 0, m and M can be taken inﬁnitesimally small, so that for some δ ' 0, F(x + dx)−F(x) = f(x)dx + δ•dx, which gives the desired result.
To show the last part of the theorem, note that for some constant c, F(x) = G(x) + c, a ≤ x ≤ b, and that F(a) = 0. The proofs of the next two theorems are not particularly of a nonstandard nature, and for that reason are kept rather short.
123
Theorem 3.7.3 (Substitution rule.) Let F0(x) = f(x) for x ∈ [a,b], and let x = g(w) for w ∈ [α,β] such that g maps [α,β] onto [a,b], with g(α) = a, g(β) = b. Also assume that g is continuously diﬀerentiable. Then, Zb a f(x)dx =Zβ α f(g(w))g0(w)dw.
Proof: From Theorem 3.6.1 it follows that,
(F(g(w)))0 = F0(g(w))g0(w) = f(g(w))g0(w),
hence, Zb a
f(x)dx = F(b)−F(a) = F(g(β))−F(g(α)) =Zβ α
(F(g(w)))0dw =
Zβ α
f(g(w))g0(w)dw.
Theorem 3.7.4 (Integration by parts.) If both f and g are continuously diﬀerentiable in [a,b], then, Zb a f(x)g0(x)dx = f(b)g(b)−f(a)g(a)−Zb a f0(x)g(x)dx.
Proof: Since (f(x)g(x))0 = f(x)g0(x) + f0(x)g(x) it follows that, J =Zb a (f(x)g(x))0dx, exists and is equal to the sum of the two integrals in the statement of the theorem. But J = f(b)g(b)−f(a)g(a).
3.8 Pitfalls in nonstandard analysis
In this section it is shown by means of a number of examples that some care is required when applying nonstandard analysis.
124
A) The existence of a positive inﬁnitesimal ε means that, ∃ε ∈∗IR : ∀m ∈ IN : 0 < ε < 1/m. By transfer (?) this would give, ? ∃ε ∈ IR : ∀m ∈ IN : 0 < ε < 1/m ?, which obviously is not true. The cause of the trouble is that in the ﬁrst statement the constant IN is external, so that transfer is not allowed.
B) Any nonempty subset X of IN has a smallest element, that is,
∃X ⊆ IN,X 6= ∅ : ∃x ∈ X : ∀y ∈ X : x ≤ y, which by transfer (?) would lead to, ? ∀X ⊆∗IN,X 6= ∅ : ∃x ∈ X : ∀y ∈ X : x ≤ y ?, which is wrong as can be seen be seen by taking, for example, X = ∗IN−IN (which is external), for if x were the smallest element of X, then also x−1 ∈ X, so x−1 ≤ x. The correct procedure would be:
∀X ∈P(IN),X 6= ∅ : ∃x ∈ X : ∀y ∈ X : x ≤ y, which by transfer gives, ∀X ∈∗(P(IN)),X 6= ∅ : ∃x ∈ X : ∀y ∈ X : x ≤ y, so that X must be internal. The latter is indeed true as could be shown by returning to ﬁrst principles, i.e. to write ∗(P(IN)) as {H(X1,X2,...) : Xi ⊆ IN}, so that, if xi is the smallest element of Xi, H(x1,x2,...) is the smallest element of H(X1,X2,...). C) As is well-known IR has Archimedian order, that is, given any x ∈ IR there is an n ∈ IN such that n > x, or, ∀x ∈ IR : ∃n ∈ IN : n > x. But ∗IR has no such order, ? ∀x ∈∗IR : ∃n ∈ IN : n > x ?, for take any x ∼∞. Indeed, transfer would give, ∀x ∈∗IR : ∃n ∈∗IN : n > x, which is correct. One might say that ∗IR has hyper-Archimedean order.
125
D) In Section 1.4 it was shown by means of transfer that statements (1.2) and (1.3) are equivalent: ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,| x−c |< δ :|∗f(x)−∗ f(c) |< ε and, ∀δ ∈∗IR,δ ' 0 : ∗f(c + δ)−∗f(c) ' 0, where c ∈ IR and f : IR → IR. Note that ∗c = c. Replacing c by a nonstandard constant or replacing ∗f by a nonstandard, but internal, function, this equivalence may be destroyed. Examples were already given in Section 3.3.
E) Let S be a bounded subset of IR, then S has a least upper bound in IR. Hence, by transfer (?), the set of all inﬁnitesimals in ∗IR would have a least upper bound β in ∗IR, which it has not, because β and hence 2β would have to be inﬁnitesimals themselves, but 2β > β. Transfer is illegal here because, by Theorem 2.10.4, the inﬁnitesimals form an external set.
126
Chapter 4
Some special topics
4.1 Principles of permanence
In the proof of Theorem 3.3.2 the fact was used that an external set is not internal, and in the remark below that proof that this fact, which is called Cauchy’s principle, is a principle of permanence. In general the latter is the statement that if a certain property P holds for all elements of a certain set A, it must hold for at least one element not in A: [∀a ∈ A : P(a)] → [∃b 6∈ A : P(b)]. The statement is based on the fact that an incompatibility between the property and the set would exist in case the property would only hold for the elements of the set. For example, let the set be IN and let the property be being an element of some given internal subset S of ∗IN: [∀a ∈ IN : a ∈ S] ⇒ [∃b 6∈ IN : b ∈ S]. If S would only contain the elements of IN there would be an incompatibility as S is internal and IN is external. In other words S must contain some ω ∼ ∞. The aspect of permanence here lies in the fact that belonging to S is necessarily carried over from the classical natural numbers to certain hyperlarge numbers. From this point of view Cauchy’s principle would not seem to be a very explicit example of a principle of permanence, and indeed some authors restrict the term ‘principle of permanence’ to cases where if something is true for all elements of some set, it must be true for some element outside that set. On the other hand all principles of permanence diﬀerent from Cauchy’s principle can be based on the latter (even Fehrele’s principle to be discussed below). It would be wrong to conclude that in the example S would contain all ω ∼∞, since taking ωo ∼∞ arbitrarily, S deﬁned by, S = {n : n ∈∗IN,n ≤ ωo},
127
128
is internal. The latter is easily shown by returning to the basic theory, because, letting ωo = H(noi), S = {H(ni) : ni ≤ noi} = {H(ni) : ni ∈ Si}, where Si = {n : 1 ≤ n ≤ noi}. Note that the example is a special case of overﬂow (see Theorem 2.11.1), and that overﬂow, as well as underﬂow, may be seen as principles of permanence. Theorem 4.1.1 If f : X → ∗IR is an internal function such that ∀x ∈ X : f(x) ' 0, then supx∈X | f(x) |' 0.
Proof 1: Since f(x) is bounded over X, the supremum exists. Denote it by β. Then ∀δ > 0 : ∃x ∈ X :| f(x) |≥ β −δ, that is β ≤ δ+ | f(x) |. Let δ ∼ 0, then it follows that β ' 0. Proof 2: Let I = {β ∈ ∗IR : [∀x ∈ X :| f(x) |≤ β]}. Then I is internal, but it contains the external subset {β ∈ ∗IR : β > 0, β is not an inﬁnitesimal}, hence, by Cauchy’s principle, it must contain some β ∼ 0, from which the result follows.
Note that the ﬁrst proof, which is classical in nature, is to be preferred. Nevertheless, the second proof is a good illustration of applying Cauchy’s principle. Corollary 4.1.1 Let [a,b] be some interval of ∗IR, such that b−a is limited, and let f and g both be Riemann integrable functions over [a,b] such that f(x) ' g(x) for all x ∈ [a,b]. Then, Zb a f(x)dx 'Zb a g(x)dx.
Proof: Let β = supa≤x≤b | f(x)−g(x) |. According to the theorem β ' 0, hence, 0 ≤

Zb a f(x)dx−Zb a g(x)dx

≤Zb a | f(x)−g(x) | dx ≤Zb a β •dx = β(b−a), which is an inﬁnitesimal.
In Cauchy’s principle an external set is ‘confronted’ with an internal set, but there is another principle of permanence (in the more general sense of the term) where two external sets that are of diﬀerent kinds are ‘confronted’ with each other. A typical example of an external set of the ﬁrst kind is the set of all inﬁnitesimals in ∗IR, and a typical example of an external set of the second kind is the set of
129
all limited numbers in ∗IR. Obviously, one must ﬁnd a general rule from which it follows that these two sets are indeed of a diﬀerent kind. An external set is of the ﬁrst kind if it is a halo, and it is of the second kind if it is a galaxy. The notions ‘halo’ and ‘galaxy’ are deﬁned below, and the second principle of permanence is Fehrele’s principle:
No halo is a galaxy, hence no galaxy is a halo. A set H is called a halo if there exists an ω ∈∗IN, ω ∼∞, such that, 1) H = ∩n∈INS(n), where (S(n)), n ∈ [1,ω]) is a hyperﬁnite internal sequence of internal subsets S(n) of some given standard set (such as ∗IN or ∗IR or something totally diﬀerent), and, 2) H is external.
The second requirement is not superﬂuous, for take all S(n) equal to some ﬁxed internal set.
Obviously, the sequence involved may be an inﬁnite sequence, because for any ω ∼ ∞ it contains a hyperﬁnite sequence with [1,ω] as its domain. It is also allowed to deﬁne S(n) for n ∈ IN only, even though the sequence would then be external. For, with the operator H as in the basic theory, let S(n) = H(Si(n)), n ∈ IN and deﬁne T(n) for n ∈∗IN by, T(H(ni)) = H(Si(ni)), so that T = H(Si) and T is internal. Moreover, T(n) = H(Si(n)) = S(n), n ∈ IN, hence T extends S as a function with domain IN to a function with domain ∗IN. It is no restriction to assume that the sequence (S(n), n ∈ [1,ω]) is nonincreasing, i.e. that S(1) ⊇ S(2) ⊇ ... ⊇ S(ω), for if this is not the case, let, S0(n) = ∩k≤nS(k),n ∈ [1,ω], then, (S0(n),n ∈ [1,ω]) is nonincreasing, and H = ∩n∈INS0(n). It may even be assumed that the sequence is strictly decreasing, i.e. that S(1) ⊃ S(2) ⊃ ... ⊃ S(ω) (perhaps for some other ω). For let, K = {n ∈ [1,ω] : [∃p(n) > n : S(n) ⊃ S(p(n))]}, then IN ⊆ K, because otherwise ∃n0 ∈ IN : ∀p > n0 : S(n0) = S(p), hence H = S(n0), but H is external and S(n0) is internal. Moreover, by the internal deﬁnition principle, K is internal, hence ∃ω0 ∼∞ : ω0 ∈ K, i.e. [1,ω0] ⊆ K. Now let m1 = 1, m2 = p(m1),..., and S00(n) = S(mn), n ∈ [1,ω0], then, (S00(n),n ∈ [1,ω0]) is strictly decreasing, and H = ∩n∈INS00(n).
130
Conversely, if H = ∩n∈INS(n) and (S(n), n ∈ [1,ω]), where ω ∼ ∞, is strictly decreasing, then H is automatically external, for let M = {n ∈∗IN : H ⊂ S(n)}. If H were internal, then, by the internal deﬁnition principle, M would be internal as well, but M = IN and IN is external. To see that M = IN observe that IN ⊆ M and suppose that H ⊂ S(ω) for some ω ∼ ∞, then H ⊂ S(ω) ⊂ S(n) for all n ∈ IN, so that H ⊂ S(ω) ⊆ ∩n∈INS(n) = H, a contradiction. This proves the next theorem.
Theorem 4.1.2 H is a halo if and only if H = ∩n∈INS(n), where (S(n), n ∈ [1,ω]) for some ω ∼∞ is a strictly decreasing internal sequence of internal sets S(n).
A set G is called a galaxy if there exists an ω ∈∗IN, ω ∼∞, such that, 1) G = ∪n∈INT(n), where (T(n), n ∈ [1,ω]) is a hyperﬁnite internal sequence of internal subsets T(n) of some given standard set, and, 2) G is external.
As before the sequence may be an inﬁnite internal sequence or be deﬁned for n ∈ IN only. Also the sequence may assumed to be nondecreasing and even strictly increasing, as can be seen by an argument similar to the one leading to the preceding theorem.
Theorem 4.1.3 G is a galaxy if and only if G = ∪n∈INT(n), where (T(n), n ∈ [1,ω]) for some ω ∼∞ is a strictly increasing internal sequence of internal sets T(n).
Remarks:
1. The given standard set is arbitrary, and hence may be some abstract set. This shows the generality of the two deﬁnitions, where numbers only play a part in the deﬁnitions of the two notions, and even this can be weakened, as can be seen from the next remark. 2. The deﬁnitions can be generalized by replacing ∗IN by some standard index set ∗A, and letting S and T be internal functions mapping the a ∗A to internal sets S(a) and T(a). Then H = ∩a∈AS(a) and G = ∪a∈AT(a).
Theorem 4.1.4 (Fehrele’s principle.) No halo is a galaxy, hence no galaxy is a halo.
131
Proof: (Van den Berg.) Assume to the contrary that some halo H is equal to some galaxy G. Let H = ∩n∈INS(n) and G = ∪n∈INT(n), with S(n) and T(n) internal, (S(n)) nonincreasing and (T(n)) nondecreasing. Then T(n) ⊆ S(n) for all n ∈ IN. Let I = {n ∈ ∗IN : T(n) ⊆ S(n)}. Then I is internal, as follows from the internal deﬁnition principle. Also IN ⊆ I, so that, since IN is external, ω ∈ I for some ω ∼ ∞, so that S(n) ⊇ S(ω) ⊇ T(ω) ⊇ T(n) for all n ∈ IN, hence H ⊇ S(ω) ⊇ T(ω) ⊇ G, or H = S(ω) = T(ω) = G, so that H = G would be internal.
Exercise: Show that the subset G of an internal set S is a galaxy if and only if S −G is a halo.
Corollary 4.1.2 (Robinson’s lemma.) Let (s(n), n ∈ ∗IN), s(n) ∈ ∗IR, be an internal sequence, such that s(n) ' 0 for all n ∈ IN. Then, ∃ω ∈∗IN,ω ∼∞ : ∀k ∈∗IN,k ≤ ω : s(k) ' 0.
Proof 1: (Van den Berg.) Let H = {n ∈ ∗IN : [∀k ≤ n : s(k) ' 0]} and G = IN. Then G is a galaxy and H ⊇ IN = G. If H is external, then G ⊂ H (by Fehrele’s principle), and if H is internal then trivially G ⊂ H, hence G ⊂ H anyway. Let ω ∈ H −G, then ω ∼∞, and s(k) ' 0 for all k ≤ ω. Proof 2: (Robinson; in time preceding the ﬁrst proof, and using Cauchy’s principle.) Let S = {n ∈ ∗IN : [∀k ≤ n :| s(k) |≤ 1/k]}, then S is internal and S ⊇ IN, hence (by Cauchy’s principle) S ⊃ IN, so that, ∃ω ∈∗IN,ω ∼∞ : ∀k ≤ ω :| s(k) |≤ 1/k. Let k ≤ ω. If k ∼ ∞, then s(k) ' 0, as then 1/k ' 0, and if k ∈ IN, then by assumption s(k) ' 0.
Corollary 4.1.3 (Dominated approximation.) Let f, g and h be functions from ∗IR to ∗IR, Riemann integrable over (−∞,+∞). Let f and g be internal but h be standard. Assume that f(x) ' g(x) for all limited x, and that | f(x) |, | g(x) |≤ h(x) for all x ∈∗IR. Then, Z+∞ −∞ f(x)dx 'Z+∞ −∞ g(x)dx.
132
Proof: Given any n ∈ IN, let β = sup |x|≤n | f(x)−g(x) |, then β ' 0, as follows from Theorem 4.1.1, and Corollary 4.1.1 implies that, ∀n ∈ IN :Z+n −n f(x)dx 'Z+n −n g(x)dx, so that, by Robinson’s lemma, there exists an ω ∼∞ such that, Z+ω −ω f(x)dx 'Z+ω −ω g(x)dx. Since, as h is standard,R|x|≥ω h(x)dx ' 0, it follows that, Z|x|≥ω f(x)dx ' 0 and Z+ω −ω f(x)dx 'Z+∞ −∞ f(x)dx, and similarly for g instead of f, from which the desired result follows. Exercise: Show that if (s(n), n ∈ ∗IN), s(n) ∈ ∗IR, is an internal sequence such that ∀n ∈ IN : s(n) |< 1/n, then ∃ω ∼∞ : ∀n ∈ [1,ω] : s(n) |< 1/n.
4.2 The saturation principle
The saturation principle is concerned with inﬁnite sequences of internal sets and does not hold in classical mathematics. An inﬁnite sequence (S(n), n ∈ IN) of sets – internal or not – has the ﬁnite intersection property if ∩n k=1S(k) 6= ∅ for alln ∈ IN.
Theorem 4.2.1 (Saturation.) Let the inﬁnite sequence (S(n), n ∈ IN) of internal sets S(n) have the ﬁnite intersection property, then the intersection of all of them is nonempty, i.e.∩n∈INS(n) 6= ∅.
Proof 1: (Not using permanence.) Let S(n) = H(Si(n)), where H is the Hoperator of the basic theory. For all n ∈ IN, let, T(n) = ∩n k=1S(k), and Ti(n) = ∩n k=1Si(k), then for n ≥ 2, T(1) ⊇ T(2) ⊇ ... ⊇ T(n), hence, {i : Ti(1) ⊇ Ti(2) ⊇ ... ⊇ Ti(n)}∈ U,
133
where U is the basic free ultraﬁlter. Also {i : Ti(n) 6= ∅∈ U}, and since i : i ≤ n} is a ﬁnite set, Qn = {i : i ≥ n,Ti(1) ⊇ Ti(2) ⊇ ... ⊇ Ti(n),Ti(n) 6= ∅}∈ U. Obviously, Qn ⊇ Qn+1. For i ∈ Q2, i ≥ 2, let ni be the maximal n ≥ 2 such that, Ti(1) ⊇ Ti(2) ⊇ ... ⊇ Ti(n), Ti(n) 6= ∅, and n ≤ i. This ni is well-deﬁned. Then {n2,n3,...} is not bounded, because if it were, so that ni ≤ m for some m ∈ IN, then Qm+1 = ∅, but Qm+1 ∈ U. Now for each i ∈ Q2, i ≥ 2 take si ∈ Ti(ni) and take si arbitrarily otherwise, then for each n ∈ IN, H(si) ∈ S(n), because for each n ∈ IN there is an ni ≥ n, so that as si ∈ Ti(ni), also si ∈ Ti(n) ⊆ Si(n), and H(si) ∈ S(n), as Q2 ∈ U. Proof 2: (Using Cauchy’s principle, and more elegant.) First extend the given sequence to an internal sequence as indicated below the deﬁnition of halo (this is not necessary in the ﬁrst proof). Now let Q = {n ∈∗IN : ∩n k=1S(k) 6= ∅}, then Qis internal and Q ⊇ IN, hence Q ⊃ IN and, ∃ω ∼∞ : ∩ω k=1S(k) 6= ∅, so that certainly ∩k∈IN S(k) 6= ∅.
Note that the second proof leads to a more general result, which in fact is a principle of permanence.
In classical mathematics a counterexample to the theorem is, for example, the sequence (S(n)) where S(n) = {n,n + 1,...}, n ∈ IN.
Corollary 4.2.1 Let A be a given internal set, and (S(n)) an inﬁnite sequence of internal subsets of A. If for all n ∈ IN, ∪n k=1S(k) 6= A, then ∪k∈INS(k) 6= A.Hence if the union of any ﬁnite number of S(n) does not ﬁll up A, then the union of all of them does not ﬁll up A.
Proof: The proof follows from the fact that (∪S(n))c = ∩Sc(n), where c denotes complementation with respect to A.
Corollary 4.2.2 Given an inﬁnite sequence (S(n)) of internal sets, then S = ∪k∈INS(k) is internal if and only if there exists an n ∈ IN such that S=∪n k=1S(k).
Proof: The if-part follows immediately. Conversely, if S is internal, then so are all T(n) = S −S(n), and ∩k∈INT(k) = ∅, hence there must exist an n ∈ IN such that ∩n k=1T(k) = ∅, which means that S = ∪n k=1S(k).
134
Corollary 4.2.3 Given an inﬁnite sequence (S(n)) of internals sets, then S = ∩k∈INS(k) is internal if and only if there exists an n ∈ IN such that S = ∩n k=1S(k).
Proof: By complementation from Corollary 4.2.1.
4.3 Stirling’s formula
In order to provide still more evidence that nonstandard mathematics can be a very elegant substitute for classical mathematics in this section Stirling’s formula for large factorials will be derived by nonstandard means. The argument closely follows that given in Van den Berg and Sari [27]. It takes the deﬁnition and properties of e as the base of the natural logarithm for granted, as well as those of π as the area of the unit circle, and that, Z+∞ −∞ exp(−x2)dx = √π. By deﬁnition, Γ(x) =Z∞ 0 e−ttx−1dt, x ∈∗IR,x > 0. Also the existence of this integral is taken for granted. Let ω be any positive hyperlarge element of ∗IR, so that, Γ(ω + 1) =Z∞ 0 e−ttωdt. The integrand is increasing in the interval [0,ω] and decreasing in the interval [ω,∞), for which reason the variable t is replaced by u = (t−ω)/ω, giving, Γ(ω + 1)ω−ω−1eω =Z∞ −1 e−ωu+ω log(1+u)du, so that the integrand now reaches its maximum at u = 0. It so happens that there exists a positive inﬁnitesimal δ such that the contributions of the integrand over the intervals [−1,−δ] and [+δ,∞) may be ignored, so that only the interval [−δ,+δ] need be taken into account. In other words, the ‘mass’ of the integrand is almost entirely concentrated in a hypersmall interval around zero. Instead of the inﬁnitesimal δ consider for the time being any d ∈ ∗IR, 0 < d < 1, split [−1,∞) into [−1,−d], [−d,+d] and [+d,∞) and indicate the integrals of e−ωu+ω log(1+u) over these subintervals by, Z−d −1 ,Z+d −d ,Z∞ +d , respectively.
135
1) [+d,∞). Since the second derivative of u−log(1+u) is positive, it follows from Taylor’s theorem that, u−log(1 + u) > d−log(1 + d) + (u−d)d/(1 + d), giving, after replacing in the integrand the left-hand side of this inequality by its right-hand side, and evaluating the resulting integral, that, 0 <Z∞ +d < e−ωd+ω log(1+d)(1 + d)/(ωd), if – in view of the denominator ωd−d is not an inﬁnitesimal. Since −d+ log(1 + d) < 0 it then further follows that, ∀m ∈ IN : 0 <Z∞ +d < ω−m. Let G = {d ∈ ∗IR : 0 < d < 1, d is not an inﬁnitesimal}, then since the positive inﬁnitesimals form a halo, G is a galaxy. But, H = ∩m∈IN{d ∈∗IR : 0 < d < 1,0 <Z∞ +d < ω−m, is a halo that clearly contains G, hence, by Fehrele’s principle, H must contain a positive inﬁnitesimal δ0, such that, ∀m ∈ IN : 0 <Z∞ +δ0 < ω−m. Obviously, δ0 may be replaced by a larger inﬁnitesimal. 2) [−1,−d]. Now, u−log(1 + u) > −d−log(1−d)−(u + d)d/(1−d), and since d + log(1−d) < 0 it follows similarly that, ∀m ∈ IN : 0 <Z−d −1 < ω−m, if again d is not an inﬁnitesimal, but nevertheless there must be a positive inﬁnitesimal δ00, that may be replaced by a larger one, such that, ∀m ∈ IN : 0 <Z−δ00 −δ00 < ω−m. The details are left as an exercise. Letting δ = max{δ0,δ00} it follows that, ∀m ∈ IN : 0 <Z∞ +δ +Z−δ −1 < 2ω−m, showing that the contributions of the two ‘tails’ are extremely small, which is not yet to say that they may be ignored.
136
3) [−δ,+δ]. By Taylor’s theorem, u−log(1+u) = (u2/2)/(1+θu)2, for some θ, 0 < θ < 1, so that u−log(1 + u) = u2(1 + ε0(u))/2 for some ε0(u) ' 0. Replacing u by v = u√ω then gives, for some ε(v) ' 0, √ω•Z+δ −δ =Z+δ√ω −δ√ω exp(−v2(1 + ε(v))/2)dv. Here δ is ﬁxed such that δ√ω ∼∞. Now let f(v) = exp(−v2(1+ε(v))/2), g(v) = exp(−v2/2), and h(v) = exp(−v2/4), then from Theorem 4.1.3 it follows that, Z+δ −δ ∼ ω−1/2 •Z+∞ −∞ exp(−v2/2)dv =q(2π/ω). Combining everything (in 2) it is suﬃcient to take m=1) ﬁnally leads to,
lim x→∞
Γ(x + 1) xxe−xq(2πx) = 1.
For more general results the reader may consult Van den Berg [28] and Koudjeti [29].
4.4 Nonstandard mathematics without the axiom of choice?
The preceding pages should have made it clear that nonstandard mathematics can be introduced in a way that is well-known to classical mathematicians. But logicians claim that from the point of view of logic and axiomatics when relating, say, IR to ∗IR our naive approach obscures the insight into what is really happening. They are right, but nevertheless a naive approach could well be more understandable and also more acceptable, because there is no verdict on external sets, and because the axioms can easily be grasped (not even the Zermelo-Fraenkel axioms of set theory are necessary, requiring to look at natural numbers as sets and implying the unintended fact that there must be hyperlarge natural numbers). Yet, one stumbling stone remains: the axiom of choice. Couldn’t we do without? Let us try and see what happens if the same general line of thinking is followed but the underlying free ultraﬁlter U over IN is replaced by the Fr´echet ﬁlter Fo (see Section 1.14). This means that again inﬁnite sequences of classical entities will generate entities that either are new or are identiﬁed with their
137
classical counterparts. It also means that we no longer follow ideas of Luxemburg and others, but follow instead those of Chwistek (see Section 1.9) and perhaps those of Cauchy, who does not seem to be very explicit, however, when it comes to deﬁning inﬁnitesimals. Anyway, it would have been diﬃcult for Cauchy to base his informal treatment of the inﬁnitesimals on a free ultraﬁlter U, because the free ultraﬁlter theorem (see the Appendix) was not known to him, and moreover the axiom of choice had in his time still to be ‘invented’. Chwistek is much more explicit, but does not develop anything that could be appreciated as a fully ﬂedged inﬁnitesimal calculus. Clearly, Q ∈ F0 if and only if i ∈ Q for i ≥ n for some n ∈ IN. The latter will be rephrased as ‘for i large enough’. As has been made clear in Section 1.10 in order to introduce ∗IR (now with respect to F0) it will again be necessary to consider all inﬁnite sequences of real numbers. Instead of, H(si) = H(ti) if and only if {i : si = ti}∈ U (Section 2.2), the deﬁnition of equality will be,
H(si) = H(ti) if and only if i is large enough,
and (Section 2.3), H(si) = s if and only if si = s for i large enough,si,s ∈ IR, Theorem 2.3.1 remains true, although the only-if part of the proof must be modiﬁed: If {i : Si = Ti}6∈ Fo, then either there is a subsequence (si(j), j ∈ IN) such that si(j) ∈ Si(j), but si(j) 6∈ Ti(j), or there is a subsequence with the roles of S and T reversed (or both). Assume the ﬁrst case, and take si ∈ Si arbitrarily if i is not an index of the subsequence. Then H(si) ∈ S and H(si) 6∈ T, i.e. S 6= T, and similarly for the other case.
So, again,
H(si) = {H(si) : si ∈ Si}, and Theorem 2.3.2 too remains valid.
Also the introduction of internal pairs, n-tuples in general and functions does not cause diﬃculties. But with Theorem 2.4.2 the problems begin. Let S = {0,1}, then S contains q = H(0,1,0,1,...). Even though S is ﬁnite, ∗S 6= S, because q 6= 0 and q 6= 1.
138
Moreover q is not hyperlarge, so what is it as an element of ∗IN? Actually ∗S turns out to be inﬁnite, and the conclusion is that ∗S contains far to many elements, i.e. the Fr´echet ﬁlter does not operate properly and allows too much to go through. But perhaps this is not really harmful. The survey at the end of Section 2.4 reveals more trouble, however, for although ∗∅, ∗ =, ∗ ∈ and ∗∪ are equal to or equivalent to ∅, =, ∈ and ∪, respectively, this is not true for: 6=, 6∈, ∩, – and c. In fact, each of the relevant equivalences or equalities must be replaced by an implication or an inclusion in the correct direction, as the reader can ﬁnd out for her or himself. It follows that ∗ 6=, ∗ 6∈, ∗∩, ∗− and ∗c are all new relations or operations. Consequently, it follows from Section 2.6 (ignoring the implications of Section 2.5 on externality) that L oˇs’ theorem (Section 2.7) is no longer true, and that the same holds for transfer (Section 2.8). As one of the many counterexamples, consider H(Si)∗∪H(Ti) that is no longer equal to H(Si)∪T(i). Section 2.7 reveals even more trouble: although ∗∧is equivalent to∧; ∗¬, ∗∨, ∗ ⇒ and ∗ ⇔ are not equivalent to ¬, ∧, ⇒ and ⇔, respectively. Finally, let us review the quantiﬁers. Although, given Xi and ci, ∃H(xi) ∈ H(Xi) : H(xi) = H(ci) is equivalent to, H(∃xi ∈ Xi : xi = ci), the equivalence is in general invalid if the simple statement xi = ci is replaced by some other statement. Similar remarks apply to ∀. Also the deﬁnition of ∗R with R some binary relation is cumbersome, for suppose that s2iRt2i but ¬(s2i−1Rt2i−1) for all i ∈ IN, then H(siRti) is neither true nor false. Compare this to the example before with H(0,1,0,1,...). So H(siRti) is not an ordinary statement, but an awkward internal something. Yet the deﬁnitions of ∗ < and ∗ >, for example, do not cause any problems, simply because < and > are not yet deﬁned for hyperreal numbers, and the following case of transfer regarding continuity is legitimate, ∀ε ∈ IR,ε > 0 : ∃δ ∈ IR,δ > 0 : ∀x ∈ IR,| x−c |< δ :| f(x)−f(c) |< ε is equivalent to, ∀ε ∈∗IR,ε > 0 : ∃δ ∈∗IR,δ > 0 : ∀x ∈∗IR,| x−c |< δ :|∗f(x)−f(c) |< ε. The deﬁnitions of inﬁnitesimal and hyperlarge number can even be given a very simple form: ε ' 0 if and only if ε = H(δi) where (δi) converges to 0,
139
and, ω ∼∞ if and only if ω = H(ωi) where (ωi) converges to +∞ or to −∞.
Remarks:
1. It may well be that Cauchy’s informal treatment comes closer to this kind of transfer, than to transfer with respect to some free ultraﬁlter U, but we will never know for sure. 2. Compare the deﬁnition of ω ∼∞ with Corollary 2.12.1. Even the following remains true: f is continuous at c ∈ IR if and only if, ∀δ ∈∗IR,δ ' 0 : ∗f(c + δ)−f(c) ' 0. The proof of this equivalence is simple, because the latter statement is equivalent to, ∀(δi,i ∈ IN), tending to 0 : (f(c + δi)−f(c),i ∈ IN) tends to 0, which is equivalent to the continuity of f at c, and we are back at the plausible reasoning of Section 1.10. Note that Cauchy applied the simpliﬁed deﬁnition also to arbitrary c ∈ ∗IR, so that he must have used some sort of S-continuity (see Section 3.3). In fact the ε−δ deﬁnition was introduced later on by Weierstrass. The conclusion must be that with Fo instead of U nonstandard mathematics becomes a very restricted theory and nothing is left of the logician’s equivalence ideal. More seriously to the ordinary mathematician, nothing is left of entirely new mathematical models that can be studied on the basis of a free ultraﬁlter and that cannot exist in classical mathematics, but nevertheless have turned out to be of great value not only within mathematics, but also outside it (such models have not been treated in this book). On the other hand what is left in the restricted theory can be based on well-known facts (such as the equivalence of (4.1) with ordinary continuity).
140
Appendix
The proof of the theorem below is based on the axiom of choice. It is the only instance where this axiom is required for the theory of nonstandard mathematics, assuming that the corresponding classical mathematics does not require it. The axiom of choice can be shown to be equivalent to Zorn’s lemma, stating that if each totally ordered subset of a partially ordered nonempty set E has an upper bound in E with respect to the implied order, then E has at least one maximal element.
E is called partially ordered if there exists a binary relation ρ, called order relation or simply order, for some or all pairs of elements of E such that for a,b,c ∈ E, 1) if aρb and bρc then aρc, 2) if aρb and bρa then a = b, and 3) aρa for all a ∈ E. A subset G of the partially ordered E with order ρ is totally ordered with respect to ρ if aρb or bρa or both for all pairs (a,b), a,b ∈ E, and m is a maximal element of E if [∀a ∈ E : mρa] implies that aρm and hence that a = m. The proof showing the equivalence of the axiom of choice and Zorn’s lemma is by no means trivial, and can be found in several textbooks, e.g. Dunford and Schwartz [30].
Theorem Free ultraﬁlters over IN exist.
Proof: By ‘ﬁlter’ will be meant ‘ﬁlter over IN’. Let Fo be the Fr´echet ﬁlter, i.e. the set of the complements of all ﬁnite subsets of IN, so that Q ∈ Fo if and only if {n,n + 1,...}⊆ Q for some n ∈ IN. Let E be the set of all ﬁlters F such that F ⊇ Fo. Then E is nonempty and can be partially ordered by means of the order ρ, where aρb if and only if a ⊆ b, i.e. the order is set inclusion. Let G be any totally ordered subset of E. Then, B = ∪{F : F ∈ G} is an element of E, and B is an upper bound of G. Obviously B ⊇ Fo, and that B is a ﬁlter can be seen as follows.
141
142
1) IN ∈ B, as IN ∈ Fo ⊆ B. 2) ∅6∈ B, as B is the union of ﬁlters. 3) If Q ∈ B and IN ⊇ R ⊇ Q, then Q ∈ F for some F ∈ B, hence R ∈ F, so R ∈ B.4) If Q,R ∈ B, then Q ∈ F and R ∈ F0 for certain F,F0 ∈ G. As G is totally ordered F ⊆ F0 or F0 ⊆ F (or both). Assume F ⊆ F0, then Q,R ∈ F0, so Q∩R ∈ F0 ⊆ B. That B is an upper bound for G is easily seen. According to Zorn’s lemma E must contain a maximal element U. Since U ⊇ Fo, U is a free ﬁlter. U is also an ultraﬁlter. For let Q ∈ U be arbitrary. In order to show that either Q ∈ U or Qc ∈ U consider the following two cases. Case 1: Suppose ∀Q0 ∈ U : Q∩Q0 is inﬁnite. Let, V = {T : T ⊆ IN,T ⊇ Q∩Q0 for some Q0 ∈ U}. Then V is a ﬁlter. The veriﬁcation of this statement is left as an exercise. Also V ∈ E, for let Q0 = {n,n + 1,...}, and U ⊆ V . By maximality V ⊆ U. But Q ∈ V , hence Q ∈ U. Case 2: Suppose ∃Q00 ∈ U : Q∩Q00 is ﬁnite. Then ∀Q0 ∈ U : Qc ∩Q0 is inﬁnite. To see this let Q0 ∈ U be arbitrary. Since both Q0 and Q00 belong to U, also Q0∩Q00 ∈ U and is inﬁnite. Since Q∩Q0∩Q00 is ﬁnite Q0∩Q00−Q∩Q0∩Q00 is inﬁnite, i.e. Qc∩Q0∩Q00 is inﬁnite and so is Qc∩Q0. Now apply Case 1 with Qc instead of Q.
References
[1] Heath, T.L., The Works of Archimedes, with the Method of Archimedes, Dover Publications, 1912, chapter VII.
[2] Euler, L., Introduction ad Analysin Inﬁnitorum, 1748.
[3] Luxemburg, W.A.J., What is Nonstandard Analysis?, American Mathematical Monthly, 80, 1973, 38–67.
[4] Cauchy, A.L., Course d’analyse de l’´ecole royale polytechnique, 1821. [5] Lakatos, Imre, Cauchy and the Continuum: The Signiﬁcance of Non-standard Analysis for the History and Philosophy of Mathematics, The Mathematical Intelligencer, 1978, 151–161.
[6] Robinson, A., Non-standard analysis, Proceedings Royal Academy, Amsterdam, Series A, 64, 1961, 432–440. [7] Robinson, A., Nonstandard Analysis, North-Holland, 1966 (2nd revised edition in 1974, 3rd edition in 1996, Princeton University Press).
[8] Hahn, H., ¨Uber die nichtarchimedische Groszensysteme, S.-B. Wiener Akademie, Math.-Natur. Kl. 116, Abt. IIa, 1907, 601–655. [9] Skolem, T., ¨Uber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzahlbare unendlich vieler Aussagen mit ausschliesslich Zahlenvariabelen, Fund. Math. 23, 1933, 150–161, [10] Hewitt, E. Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64, 1948, 45–99. [11] L oˇs, J., Quelques remarques, theor`emes, et probl`emes sur les classes deﬁnissables d’algebras, in: Skolem et al., eds., Mathematical interpretations of formal systems, North-Holland, 1955, 98–113.
[12] Laugwitz, D., and C. Schmieden, Eine Erweiterung der Inﬁnitesimalrechnung, Mathematische Zeitschrift 69, 1958, 01–39.
[13] Luxemburg, W.A.J., Nonstandard Analysis, Lectures on A. Robinson’s theory of inﬁnitesimal and inﬁnitely large numbers, Caltech Bookstore, 1962.
[14] Nelson, E., Internal set theory, Bull. Amer. Math. Soc. 83, 1977, 1165–1193.
[15] Robert, A., Nonstandard Analysis, Wiley, 1988. [16] Diener, F., et G. Reeb, Analyse Non Standard, Hermann, 1989.
143
144
[17] Chwistek, L., ¨Uber die Hypothesen der Mengenlehre, Mathematische Zeitschrift 25, 1926, 439–473.
[18] Chwistek, L., The Limits of Science, translated from the Polish, Routledge Kegan Paul, 1948. [19] Chang, C.C., and H.J. Keisler, Model Theory, North-Holland,
[20] Beeson, M.J., Foundations of Constructive Mathematics, Springer, 1985.
[21] Bishop, E., and D. Bridges, Constructive Analysis, Springer, 1985.
[22] Beth, E.W., The Foundation of Mathematics, North-Holland, 1965. [23] Beth, E.W., Mathematical Thought, Reidel, 1965, chapter V.
[24] Heyting, A., Intuitionism, An Introduction, North-Holland, 1966. [25] Potter, M.D., Sets: An Introduction, Clarendon Press, 1990.
[26] Keisler, H.J., Elementary Calculus, An Inﬁnitesimal Approach, Prindle, Weber Schmidt, 1986.
[27] Berg, I.P. van den, en T. Sari, Inleiding tot de inﬁnitesimaalrekening, (Introduction to the inﬁnitesimal calculus; Lecture Notes, University of Groningen, 1988; private communication; in Dutch).
[28] Berg, I.P. van den, Nonstandard Asymptotic Analysis, Lecture Notes in Mathematics, nr. 1249, Springer, 1987. [29] Koudjeti, F., Elements of External Calculus with an Application to Mathematical Finance, thesis, University of Groningen, 1995.
[30] Dunford, N., J.T. Schwartz, Linear Operators, part I, Interscience, 1957.
Index
accumulation point, 111 appreciable, 18, 83, 96 Archimedian order, 122 ∗-transform, 19, 28, 95 ∗-transform expression, 69 ∗-transform of relation, 71 ∗-transform operation, 69 ∗-transforms of attributes, 81 atomic relation, 26, 71 atomic statement, 71 axiom of choice, 15, 34, 40, 45, 139 axiomatics, 134 axioms of set theory, 15
basic assumptions regarding n-tuples, 48 basicassumptionsregardinggeneratingnew constants, 49 basic assumptions regarding logic, 45 basic assumptions regarding sets, 46 basicassumptionsregardingthenaturalnumbers, 45 bijective, 50 bound variable, 21 boundary point, 111 bounded set, 109
Cartesian product, 50 Cauchy sequence, 55, 91 Cauchy’s principle, 106, 125, 131 chain rule, 115 classical mathematics, 15 closed set, 110 closure, 111 complement of subset, 47 composite function theorem, 104 concurrent Cauchy sequences, 56 constant, 46 constructivism, 41, 43 constructivistic mathematics, 15 continuity, 82, 100, 104 countable, 82
countably inﬁnite set, 50 critical point theorem, 116
denumerably inﬁnite set, 50 diﬀerence of sets, 70 diﬀerentiation, 114 direct product, 50 domain, 50, 112 dominated approximation, 129 dummy variable, 21
elements of set, 47 empty set, 47 equality, 47 equality of integers, 55 equality of internal constants, 58 equality of rationals, 55 equality of reals, 55 excluded third, 15 exhaustion, 30 existence, 46 existential quantiﬁer, 20 expressions, 69 extensionality, 47 external constant, 28, 57, 67 external notion, 94 external set, 134 extreme value theorem, 103
Fehrele’s principle, 128, 133 ﬁlter, 51 ﬁnite, 18, 83 ﬁnite intersection property, 130 ﬁnite number, 73 ﬁnite sequence, 50 ﬁnite set, 50, 81 formalism, 41 formalistic mathematics, 15 formulae, 73 Fr´echet ﬁlter, 52, 134, 139 free ultraﬁlter, 40, 49, 52, 53, 134, 139
145
146
free variable, 20 function, 49 function value, 50
galaxy, 127 generalized mean value theorem, 117 generating sequence, 94 generation inﬁnite sequences, 134 graph, 49
halo, 110, 127 hyper n-tuple, 61 hyper-Archimedean order, 122 hyperconstant, 57 hypercontinuity, 82 hypercountable, 82 hyperﬁnite number, 82 hyperﬁnite set, 81 hyperfunction, 58, 62 hyperlarge, 17 hypernumber, 58 hyperpair, 58 hyperreal, 18 hyperreal number, 82 hyperset, 58 hypersmall, 87 hypersmall number, 17
idealization, 35 identiﬁcation, 49 identiﬁcation of integers, 55 identiﬁcation of internal n-tuples, 61 identiﬁcation of internal constants, 58 identiﬁcation of internal functions, 61 identiﬁcation of internal sequences, 62 identiﬁcation of internal sets, 59 identiﬁcation of rationals, 55 identiﬁcation of reals, 55 improper integral, 119 increment theorem, 115 individual, 47 inductive proof, 46 inﬁnite sequence, 50, 102 inﬁnite set, 50 inﬁnitely large, 16, 17, 83 inﬁnitely large number, 96 inﬁnitesimal, 15, 17, 83, 96 injective, 50 integers, 55 integration, 118
integration by parts, 118 interior point, 110 intermediate value theorem, 103 internal n-tuples, 61 internal composition, 62 internal constant, 28, 57, 63 internal deﬁnition principle, 71, 96 internal function, 61, 104 internal notion, 94 internal pair, 61 internal sequence, 62 internal set theory, 57 intuitionism, 43 inverse function, 112 inverse of a bijective function, 50 inverse overﬂow, 86, 97 inverse underﬂow, 86, 97
L’Hopital’s theorem, 117 least upper bound theorem, 81, 99 limit, 102, 104 limit deﬁnition, 15 limited, 18, 83 logic, 134 logical connective, 20 L oˇs’ theorem, 72
map, 49 mapping, 49 maximal element, 139 mean value theorem, 117 monotonically decreasing, 112 monotonically increasing, 112
n-tuple, 46, 94 natural extension, 67 natural number, 15 negative hyperlarge, 83 nonstandard analysis, 15 nonstandard constant, 57 nonstandard number, 16 nonstandard version, 67 number, 94
one-to-one, 50 one-to-one onto, 50 onto, 50 open set, 110 operation, 69 ordered pair, 48
147
overﬂow, 85, 97 overspill, 85, 97
pair, 15, 46 paradoxes, 48 partial order, 139 permanence, 130 positive hyperlarge, 83 power set, 47, 68, 70 predicate, 73 prenex normal form, 22 primitive notion, 15 principle of permanence, 106, 125
range, 50, 112 rationals, 55 real number, 82 real number system, 15 reals, 55 recursive functions, 43 regular of level k, 47 relation, 71 Riemann integral, 118 Riemann integration, 118 Riemann sum, 119 Robinson’s lemma, 129 Rolle’s theorem, 116
S-continuity, 105, 137 S-uniform continuity, 106 S-diﬀerentiable, 115 S-limit, 106 saturation, 130 sentence, 73 sequence, 50 set, 15, 46, 94 speciﬁcation, 46 standard constant, 34 standard copy, 66 standard deﬁnition principle, 81, 96 standard notion, 94 standard part, 19, 91, 98 standardization, 35 statement, 96 Stirling’s formula, 132 subset, 47 substitution rule, 121 surjective, 50
Taylor’s theorem, 117

珍贵数学文献（II）相关推荐

数学故事中的「蛋」和「母鸡」
本文作者:殷堰工,江苏省中职首批正高级讲师.苏州大学硕士生导师.苏州科技大学兼职教授.曾结集出版个人已发表的论文 20 万字的<数学解题策略精编>(上海科技教育出版社,1994)及发表过的 ...
机器学习中的数学：一份新鲜出炉的热门草稿
来源:机器之心本文约1500字,建议阅读5分钟. 本文为你分享近日<Mathematics for Machine Learning>的全部草稿已放出,我们整理了这本书的简要概述. 近日 ...
叶老师的对于数学的分析的摘抄！十分的认可！
进了大学,碰到了一位恩师,他让我懂得数学并不是像想象中的那么死板,生搬硬套,原来通过简单的一行公式,一个图形或者一个命题,就可以把看似复杂的现象表达得凌厉尽致; 他让我开始懂得欣赏数学.再就是从中国大 ...
开源数学软件zzllrr Mather小乐数学简介Summary
中文摘要 Summary in Chinese 基于网页编程开发经验,和对数学学科的持久热爱及钻研精神,原创开源实现一款跨平台可离线运行并满足多用户背景.多场景需要的数学专业软件:小乐数学zzllrr ...
视觉开发需要什么程度的数学_角度的概念在视觉上非常直观，但其数学定义并不是那么简单...
角的概念是几何学中最基本的概念之一.当我们研究三角形的性质时,我们自然地建立了三角形的边和角之间的联系.这些联系是在三角学中系统地建立起来的. 角是什么?我们如何测量它? 虽然角度的概念在视觉上很直观 ...
小人物解决四大数学问题：记传奇华人数学家李天岩
撰文 | 丁玖(南密西西比大学数学系教授) 责编 | 黄俊如 ● ● ● 这个月是华人数学家李天岩的74周岁华诞,也是他自密歇根州立大学以 "全校级杰出教授"(University ...
学术-数学：黎曼猜想
ylbtech-学术-数学:黎曼猜想黎曼假设(或称黎曼猜想)是关于黎曼ζ函数ζ(s)的零点分布的猜想,由数学家波恩哈德·黎曼于1859年提出.德国数学家戴维·希尔伯特在第二届国际数学家大会上提出了2 ...
[转]谈谈工科学生如何学习数学
不少工科学生特别是工科研究生对数学基础不足感到压力.确实,缺乏数学的帮助会使得学生们的研究缺乏思路和工具,缺乏捕捉问题的敏感性,缺乏抽取问题本质的能力,缺乏处理问题的技巧和方法.我们许多硕士生.博士生 ...
拒绝时间与脑力浪费，不做数学守旧派
当前,全国普通高校高等数学教材中,存在大量时间及脑力浪费现象. 1960年深秋,在数学证明中,美国数学家鲁滨逊提出两种方法:(1)全新的非标准方法(2)传统的标准方法.大量实例证明,在数学证明过程中, ...

珍贵数学文献（II）

珍贵数学文献（II）相关推荐

最新文章

热门文章