内集理论是处理非标准分析的新方法

 1977年，Nenlson（1932-2014）给自己的

代表作起了这个古怪的名字。原文如下：

Nenlson E., “Internal set theory. A new approach to nonstandard analysis,” Bull. Amer. Math. Soc., 83, No. 6, 1165–1198 ( 1977).

大家春节好!

袁萌陈启清 1月23日

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 Internal Set Theory1

 Ordinarily in mathematics, when one introduces a new concept one deﬁnes it. For example, if this were a book on “blobs” I would begin with a deﬁnition of this new predicate: x is a blob in case x is a topological space such that no uncountable subset is Hausdorﬀ. Then we would be all set to study blobs. Fortunately, this is not a book about blobs, and I want to do something diﬀerent. I want to begin by introducing a new predicate “standard” to ordinary mathematics without deﬁning it. The reason for not deﬁning “standard” is that it ys a syntactical, rather than a semantic, role in the theory. It is similar to the use of “ﬁxed” in informal mathematical discourse. One does not deﬁne this notion, nor consider the set of all ﬁxed natural numbers. The statement “there is a natural number bigger than any ﬁxed natural number” does not appear paradoxical. The predicate “standard” will be used in much the same way, so that we shall assert “there is a natural number bigger than any standard natural number.” But the predicate “standard”— unlike “ﬁxed”—will be part of the formal language of our theory, and this will allow us to take the further step of saying, “call such a natural number, one that is bigger than any standard natural number, unlimited.” We shall introduce axioms for handling this new predicate “standard” in a consistent way. In doing so, we do not enlarge the world of mathematical objects in any way, we merely construct a richer language to discuss the same objects as before. In this way we construct a theory extending ordinary mathematics, called Internal Set Theory1 that axiomatizes a portion of Abraham Robinson’s nonstandard analysis. In this construction, nothing in ordinary mathematics is changed. 1It was ﬁrst presented in [Ne] Edward Nelson, “Internal set theory: A new approach to nonstandard analysis,” Bulletin American Mathematical Society 83 (1977), 1165–1198. A good introductory account is [Rt] Alain Robert, “Analyse non standard,” Presses polytechniques romandes, EPFL Centre Midi, CH–1015 Lausanne, 1985; translated by the author as “Nonstandard Analysis,” Wiley, New York, 1988.

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 2 CHAPTER 1. INTERNAL SET THEORY

 1.1 External concepts

 Let us begin by adjoining to ordinary mathematics a new predicate called standard. Using this new predicate, we can introduce new notions. For example, we can make the following deﬁnitions, in which the variables range over R, the set of all real numbers: x is inﬁnitesimal in case for all standard ε > 0 we have|x|≤ ε, x is limited in case for some standard r we have |x|≤ r, x ' y (x is inﬁnitely close to y) in case x−y is inﬁnitesimal, x y (x is strongly less than y) in case x < y and x 6' y, x y (x is strongly greater than y) in case y x, x '∞ in case x ≥0 and x is unlimited (i.e., not limited), x '−∞ in case −x '∞, x ∞ in case x 6'∞, x −∞ in case x 6'−∞, x < ∼ y (x is nearly less than y) in case x ≤ y + α for some inﬁnitesimal α, x > ∼ y (x is nearly greater than y) in case y < ∼ x. A formula of ordinary mathematics—one that does not involve the new predicate “standard”, even indirectly—is called internal; otherwise, a formula is called external. The eleven deﬁnitions given above are all external formulas. Notice that the adjectives “internal” and “external” are metamathematical concepts—they are properties of formulas and do not apply to sets. One of the basic principles of ordinary mathematics, or internal mathematics as I shall call it from now on, is the subset axiom. This asserts that if X is a set then there is a set S, denoted by {x ∈ X : A}, such that for all x we have x ∈ S ↔ x ∈ X & A. Here A is a formula of internal mathematics. Usually it will have x as a free variable, but it may have other free variables as well. When we want to emphasize its dependence on x, we write it as A(x). Nothing in internal mathematics has been changed by introducing the new predicate, so the subset axiom continues to hold. But nothing in internal mathematics refers to the new predicate, so nothing entitles us to apply the subset axiom to external formulas. For example, we cannot prove that there exists a set I such that x ∈ I ↔ x ∈R & x is inﬁnitesimal. The notation {x ∈R : x '0} is not allowed; it is an example of illegal set formation. Only internal

 §1.2 THE TRANSFER PRINCIPLE 3 formulas can be used to deﬁne subsets. (Nevertheless, in Chapter 4 we shall ﬁnd a way to introduce so-called external sets.) Certain set formations that might at ﬁrst sight appear to be illegal are perfectly legitimate. For example, suppose that we have an inﬁnitesimal x > 0. Then we can form the closed interval [−x,x] consisting of all y such that −x ≤ y ≤ x. This is simply because we already know that for any x > 0 we can form the set [−x,x]. Exercises for Section 1.1 1. Let n be a nonstandard natural number. Can one form the set of all natural numbers k such that k ≤ n? Is this set ﬁnite? 2. Can one form the set of all limited real numbers? 3. Can one prove that every standard positive real number is limited? 4. Can one form the set of all standard real numbers x such that x2 ≤ 1? 5. Assume that 1 is standard. Can one form the set of all limited real numbers x such that x2 ≤ 1? 6. Can one prove that the sum of two inﬁnitesimals is inﬁnitesimal?

 1.2 The transfer principle

 We cannot yet prove anything of interest involving “standard” because we have made no assumptions about it. Our ﬁrst axiom is the transfer principle (T). The notation ∀st means “for all standard”, and ∃st means “there existsastandard”. LetAbeaninternalformulawhoseonlyfreevariables are x, t1, ..., tn. Then the transfer principle is ∀stt1•••∀sttn[∀stxA↔∀xA]. (1.1) We may think of the t1, ..., tn as parameters; we are mainly interested in x. Then the transfer principle asserts that if we have an internal formula A, and all the parameters have standard values, and if we know that A holds for all standard x, then it holds for all x. (The converse direction is trivial, and we could have stated (T) with just → instead of ↔.)The intuition behind (T) is that if something is true for a ﬁxed, but arbitrary, x then it is true for all x. Notice that two formulas are equivalent if and only if their negations are. But we have ¬∀x¬A ↔ ∃xA, so if we apply (T) to ¬A we obtain the dual form of the transfer principle: ∀stt1•••∀sttn[∃stxA↔∃xA]. (1.2)

 4 CHAPTER 1. INTERNAL SET THEORY Let us write A ≡ B (A is weakly equivalent to B) to mean that for all standard values of the free variables in the formulas, we have A↔B. Then we can rewrite the two forms of (T) as ∀stxA≡∀xA and ∃stxA ≡ ∃xA whenever A is an internal formula. Applying these rules repeatedly, we see that any internal formula A is weakly equivalent to the formula Ast obtained by replacing each ∀ by ∀st and ∃ by ∃st. Then t1,...,tn are standard →Ast, where t1, ..., tn are the free variables in A, is called the relativization of A to the standard sets. Consider an object, such as the empty set∅, the natural numbers N, or the real numbers R, that can be described uniquely within internal mathematics. That is, suppose that there is an internal formula A(x) whose only free variable is x such that we can prove existence ∃xA(x) and uniqueness A(x1) & A(x2)→ x1 = x2. By the dual form of transfer, ∃stxA(x); so by uniqueness, the x such that A(x) holds is standard. For example, let A(x) be ∀y[y / ∈ x]. There is a unique set, the empty set ∅,that satisﬁes A( x). Therefore ∅ is standard. The formulas describing Nand R are longer, but by the same reasoning, N and R are standard. Any object that can be uniquely described within internal mathematics is standard: the real numbers 0, 1, and π, the Hilbert space L2(R,dx) where dx is Lebesgue measure, the ﬁrst uncountable ordinal, and the loop space of the ﬁfteen dimensional sphere are all standard. The real number 10−100 is standard, so if x is inﬁnitesimal then |x|≤10−100. The same reasoning applies to internal formulas A(x) containing parameters—provided the parameters have standard values. For example, if t is a standard real number then so is sint (let A(x,t) be x = sint) and if X is a standard Banach space so is its dual. As an example of transfer, we know that for all real x > 0 there is a natural number n such that nx ≥ 1; therefore, for all standard x > 0 there is a standard n such that nx ≥ 1. But suppose that x > 0 is inﬁnitesimal. Do we know that there is a natural number n such that nx ≥ 1? Of course; we already know this for all x > 0. But if we try to argue as follows—“there is an n such that nx ≥ 1; therefore, by the dual form of transfer, there is a standard n such that nx ≥1”—then we have made an error: transfer is only valid for the standard values of the parameters (in this case x) in the formula. This is an example of illegal transfer. It is the most common error in learning nonstandard analysis. Before applying transfer, one must make sure that any parameters in the formula—even those that may be implicit in the discussion—have standard values.

 §1.3 THE IDEALIZATION PRINCIPLE 5 Another form of illegal transfer is the attempt to apply it to an external formula. For example, consider “for all standard natural numbers n, the number n is limited; by transfer, all natural numbers are limited”. This is incorrect. Before applying transfer, one must check two things: that the formula is internal and that all parameters in it have standard values.

 Exercises for Section 1.2 1. Can one prove that the sum of two inﬁnitesimals is inﬁnitesimal? 2. If r and s are limited, so are r + s and rs. 3. If x ' 0 and |r|∞, then xr ' 0. 4. If x 6= 0, then x is inﬁnitesimal if and only if 1/x is unlimited. 5. Is it true that the inﬁnitesimals are a maximal ideal in the integral domain of limited real numbers? What does the quotient ﬁeld look like? 6. Consider the function f:x 7→ x2 on a closed interval. Then f is bounded, and since it is standard it has a standard bound.—Is this reasoning correct? 7. Let x and y be standard with x < ∼ y. Prove that x ≤ y. 1.3 The idealization principle

 So far, we have no way to prove that any nonstandard objects exist. Our next assumption is the idealization principle (I). The notation ∀stﬁn means “for all standard ﬁnite sets”, and ∃stﬁn means “there exists a standard ﬁnite set such that”. Also, ∀x∈X means “for all x in X”, and∃x∈X means “there exists x in X such that”. Let A be an internal formula. Then the idealization principle is ∀stﬁnx0∃y∀x∈x0A↔∃y∀stxA. (1.3) There are no particular pitfalls connected with this assumption: we must just be sure that A is internal. It can contain free variables in addition to x and y (except for x0). The intuition behind (I) is that we can only ﬁx a ﬁnite number of objects at a time. To say that there is a y such that for all ﬁxed x we have A is the same as saying that for any ﬁxed ﬁnite set of x’s there is a y such that A holds for all of them. As a ﬁrst application of the idealization principle, let A be the formula y 6= x. Then for every ﬁnite set x0, and so in particular for every standard ﬁnite set x0, there is a y such that for all x in x0 we have y 6= x. Therefore, thereexistsanonstandard y. Thesameargumentworkswhen x and y are restrictedto range overany inﬁnite set. Inother words, every

 6 CHAPTER 1. INTERNAL SET THEORY

 inﬁnite set contains a nonstandard element. In particular, there exists a nonstandard natural number. By transfer, we know that 0 is standard, and we know that if n is standard then so is n + 1. Do we have a contradiction here? Why can’t we say that by induction, all natural numbers are standard? The induction theorem says this: if S is a subset of N such that 0 is in S and such that whenever n is in S then n + 1 is in S, then S = N. So to apply induction to prove that every natural number is standard, we would need a set S such that n is in S if and only if n is standard, and this we don’t have. As a rough rule of thumb, until one feels at ease with nonstandard analysis, it is best to apply the familiar rules of internal mathematics freely to elements, but to be careful when working with sets of elements. (From a foundational point of view, everything in mathematics is a set. For example, a real number is an equivalence class of Cauchy sequences of rational numbers. Even a natural number is a set: the number 0 is the empty set, the number 1 is the set whose only element is 0, the number 2 is the set whose only elements are 0 and 1, etc. When I refer to “elements” or “objects” rather than to sets, only a psychological distinction is intended.) We can prove that certain subsets of R and N do not exist. Theorem 1. There does not exist S1, S2, S3, S4, or S5 such that, for all n in N and x in R, we have n ∈ S1 ↔ n is standard, n ∈ S2 ↔ n is nonstandard, x ∈ S3 ↔ x is limited, x ∈ S4 ↔ x is unlimited, or x ∈ S5 ↔ x is inﬁnitesimal. Proof. As we have seen, the existence of S1 would violate the induction theorem. If S2 existed we could take S1 = N\S2, if S3 existed we could take S1 = N∩S3, if S4 existed we could take S3 = R\S4, and if S5 existed we could take S4 ={x ∈R : 1/x ∈ S5}. This may seem like a negative result, but it is frequently used in proofs. Suppose that we have shown that a certain internal property A(x) holds for every inﬁnitesimal x; then we automatically know that A(x) holds for some non-inﬁnitesimal x, for otherwise we could let S5 be the set {x ∈R : A(x)}. This is called overspill. By Theorem 1 there is a non-zero inﬁnitesimal, for otherwise we could let S5 ={0}. We can also see this directly from (I). We denote by R+ the set of all strictly positive real numbers. For every ﬁnite subset x0 of R+ there is a y in R+ such that y ≤ x for all x in x0. By (I), there is a y in R+ such that y ≤ x for all standard x in R+. That is, there exists an inﬁnitesimal y > 0.

 §1.3 THE IDEALIZATION PRINCIPLE 7 Notice that in this example, to say that y is smaller than every elementof x0 isthesameassayingthat y issmallerthantheleastelement of x0, so the ﬁnite set really plays no essential role. This situation occurs so frequently that it is worth discussing it in a general context in which the idealization principle takes a simpler form. Recall that a directed set is a set D together with a transitive binary relation ≺ such that every pair of elements in D has an upper bound. For example, R+ is a directed set with respect to ≥. (In this example, the minimum of two elements is an upper bound for them.) Let A be a formula with the free variable x restricted to range over the directed set D. We say that A ﬁlters in x (with respect to D and≺) in case one can show that whenever x ≺ z and A(z), then A(x). (Here A(z) is the formula obtained by substituting z for each free occurrence of x in A(x), with the understanding that z is not a bound variable of A.) Suppose that the internal formula A ﬁlters in x. Then the idealization principle takes the simpler form ∀stx∃yA↔∃y∀stxA. (1.4) That is, we can simply interchange the two quantiﬁers. The idealization principle also has a dual form. If A is internal, then ∃stﬁnx0∀y∃x∈x0A↔∀y∃stxA. (1.5) We say that A coﬁlters in x in case we can show that whenever x ≺ z and A(x), then A(z). Then (1.5) takes the simpler form: ∃stx∀yA↔∀y∃stxA. (1.6) In practice, there is no need to make the distinction between ﬁltering and coﬁltering, and I may say “ﬁlters” when “coﬁlters” is correct. As an example of the dual form, suppose that f:R→R is such that every value of f is limited. Then f is bounded, and even has a standard bound. To see this, use (1.6), where the ﬁltering relation is ≤ on the values of the function. We know that for all y there exists a standard x such that |f(y)|≤ x. Hence there is a standard x such that |f(y)|≤ x for all y.

 8 CHAPTER 1. INTERNAL SET THEORY

 Theorem 2. Let S be a set. Then S is a standard ﬁnite set if and only if every element of S is standard. Proof. Let S be a standard ﬁnite set, and suppose that it contains a nonstandard y. Then there exists a y in S such that for all standard x we have y 6= x, so by (I), for all standard ﬁnite sets x0, and in particular for S, there is a y in S such that for all x in S we have y 6= x. This is a contradiction, so every element of S is standard. Conversely, suppose that every element of S is standard. We already know that S must be ﬁnite, because every inﬁnite set contains a nonstandard element. Let x and y range over S and apply the dual form of idealization to the formula x = y. We know that for all y there is a standard x with x = y, so there is a standard ﬁnite set x0 such that for all y there is an x in x0 with x = y. Then x0 ⊇ S; that is, S ∈ ℘x0 where ℘ denotes the power set. But x0 is standard, so by (T) ℘x0 is standard. Also, x0 is ﬁnite, so ℘x0 is a standard ﬁnite set. By the forward direction of the theorem established in the preceding paragraph, every element of ℘x0 is standard, so S is standard.2 This has the following corollary. Theorem 3. Let n and k be natural numbers, with n standard and k ≤ n. Then k is standard. Proof. Let S = {k ∈ N : k ≤ n}. By (T), S is standard. It is ﬁnite, so all of its elements are standard. Therefore we can picture the natural numbers as lying on a tape, with the standard numbers to the left and the nonstandard numbers to the right. The demarcation between the two portions is strange: the left portion is not a set, and neither is the right. I want to emphasize that we did not start with the left portion and invent a new right portion to be tacked on to it—we started with the whole tape, the familiar set N of all natural numbers, and invented a new way of looking at it. Theorem 4. There is a ﬁnite set that contains every standard object. Proof. This is easy. Just apply (I) to the formula x ∈ y & y isﬁnite. If we think of “standard” semantically—with the world of mathematical objects spread out before us, some bearing the label “standard” and others not—then these results violate our intuition about ﬁnite sets. But recall what it means to say that X is a ﬁnite set. This is an internal 2I am grateful to Will Schneeberger for pointing out a gap in my ﬁrst proof of this theorem.

 §1.3 THE IDEALIZATION PRINCIPLE 9 notion, so it means what it has always meant in mathematics. What has it always meant? There are two equivalent characterizations. For n in N, we let In ={k ∈N : k < n}. Then a set X is ﬁnite if and only if there is a bijection of X with In for some n. There is also the Dedekind characterization: X is ﬁnite if and only if there is no bijection of X with a proper subset of itself. Consider the set In where n is a nonstandard natural number. This certainly satisﬁes the ﬁrst property. Nothing was said about n being standard; this cannot even be formulated within internal mathematics. But does it satisfy the Dedekind property? Suppose that we send each nonstandard element of In to its predecessor and leave the standard elements alone; isn’t this a bijection of In with In−1? No, its deﬁnition as a function would involve illegal set formation.—Perhaps it is fair to say that “ﬁnite” does not mean what we have always thought it to mean. What have we always thought it to mean? I used to think that I knew what I had always thought it to mean, but I no longer think so. In any case, intuition changes with experience. I ﬁnd it intuitive to think that very, very large natural numbers and very, very small strictly positive real numbers were there all along, and now we have a suitable language for discussing them.

 Exercises for Section 1.3 1. A natural number is standard if and only if it is limited. 2. Does there exist an unlimited prime? 3. What does the decimal expansion of an inﬁnitesimal look like? 4. Let H be a Hilbert space. Does there exist a ﬁnite dimensional subspace containing all of its standard elements? Is it closed? 5. Suppose that the Sn, for n in N, are a sequence of disjoint sets with union S, and suppose that every element x in S is in Sn for some standard n. Can one show that all but a ﬁnite number of them are empty? Can one show that all but a standard ﬁnite number of them are empty? 6. Prove Robinson’s lemma: Let n 7→ an be a sequence such that an ' 0 for all standard n. Then there is an unlimited N such that an ' 0 for all n ≤ N. 7. Let n 7→ an be a sequence such that an ∞ for all standard n. Is there an unlimited N such that an ∞ for all n ≤ N? 8. We have been saying “let x range over X” to mean that each ∀xA should be replaced by ∀x[x ∈ X → A] and each ∃xA should be replaced by ∃x[x ∈ X & A]. Let ℘ﬁnX be the set of all non-empty ﬁnite subsets of X. Show that if x ranges over X, then (I) can be written as ∀stx0∃y∀x∈x0A ↔∃y∀stxA where x0 ranges over ℘ﬁnX.

 10 CHAPTER 1. INTERNAL SET THEORY

 1.4 The standardization principle

 Our ﬁnal assumption about the new predicate “standard” is the standardization principle (S). It states that ∀stX∃stY∀stz[z ∈ Y ↔ z ∈ X & A]. (1.7) Here A can be any formula (not containing Y), external or internal. It may contain parameters (free variables in addition to z and X). The intuition behind (S) is that if we have a ﬁxed set, then we can specify a ﬁxed subset of it by giving a criterion for judging whether each ﬁxed element is a member of it or not. Twosetsareequaliftheyhavethesameelements. By(T),twostandard sets are equal if they have the same standard elements. Therefore, the standard set Y given by (S) is unique. It is denoted by S{z ∈ X : A}, which may be read as “the standard set whose standard elements are those standard elements of X such that A holds”. Unfortunately, any shortening of this cumbersome phrase is apt to be misleading. For standard elements z, we have a direct criterion for z to be an element of S{z ∈ X : A}, namely that A(z) hold. But for nonstandard elements z, this is not so. It may happen that z is in S{z ∈ X : A}but A(z) does not hold, and conversely A(z) may hold without z being in S{z ∈ X : A}. For example, let X = S{z ∈ R : z ' 0}. Then X = {0} since 0 is the only standard inﬁnitesimal. Thus we can have z ' 0 without z being in X. Let Y = S{z ∈R :|z|∞}. Then Y = R since every standard number is limited. Thus we can have z ∈ Y without z being limited. The standardization principle is useful in making deﬁnitions. Let x range over the standard set X. When we make a deﬁnition of the form: “for x standard, x is something-or-other in case a certain property holds”, this is understood to mean the same as “x is something-or-other in case x ∈ S{x ∈ X : a certain property holds}”. For example, let f:R → R and let us say that for f standard, f is nice in case every value of f is limited. Thus f is nice if and only if f ∈ N, where we let N = S{f ∈ RR : every value of f is limited}. (The notation XY signiﬁes the set of all functions from Y to X.) I claim that f is nice if and only if f is bounded. To prove this, we may, by (T), assume that f is standard. We already saw, in the previous section, that if every value of f is limited, then f is bounded. Conversely, let f be bounded. Then by (T) it has a standard bound, and so is nice. This proves the claim. But, one might object, the ﬁrst transfer is illegal because “nice” was deﬁned externally. The point is this: being nice is the same as being

 §1.4 THE STANDARDIZATION PRINCIPLE 11 in the standard set N, so the transfer is legal. In complete detail, (T) tells us that for all standard N we have ∀stf[f ∈ N ↔ f is bounded]→∀f[f ∈ N ↔ f is bounded]. Inotherwords, thiskindofdeﬁnitionbymeansof(S)isawayofdeﬁning, somewhat implicitly, an internal property. Let x and y range over a set V , let ˜ y range over V V , and let A be internal. Then ∀x∃yA(x,y)↔∃˜ y∀xAx, ˜ y(x). The forward direction is the axiom of choice, and the backward direction is trivial. If we apply transfer to this, assuming that V is standard, we obtain ∀stx∃styA(x,y)≡∃st˜ y∀stxAx, ˜ y(x). But we can do much better than this. Let A be any formula, external or internal. Then ∀stx∃styA(x,y)↔∃st˜ y∀stxAx, ˜ y(x). (1.8) The backward direction is trivial, so we need only consider the forward direction. Suppose ﬁrst that for all standard x there is a unique standard y such that A(x,y). Then we can let ˜ y = S{hx,yi : A(x,y)}. Inthe general case, let e Y = S{hx,Yi: Y = S{y : A(x,y)}}. Then e Y is a standard set-valued function whose values are non-empty sets, so by the axiom of choice relativized to the standard sets, it has a standard cross-section ˜ y of the desired form. We call (1.8) the functional form of standardization (eS). It has a dual form: ∃stx∀styA(x,y)↔∀st˜ y∃stxAx, ˜ y(x). (1.9)The requirements in (1.8) and (1.9) are that x and y range over some standard set V and that ˜ y range over V V . This includes the case that x and y range over diﬀerent standard sets X and Y, with ˜ y ranging over Y X, becausewecanalwayslet V = X∪Y andincludetheconditions x ∈ X and y ∈ Y in the formula A. If X is a set, contained in some standard set V , we let SX = S{x ∈ V : x ∈ X}.

 12 CHAPTER 1. INTERNAL SET THEORY

 This clearly does not depend on the choice of V , and in practice the requirement that X be contained in some standard set is not restrictive. Then SX is the unique standard set having the same standard elements as X. It is easy to see that if f is a function then so is Sf. The theory obtained by adjoining (I), (S), and (T) to internal mathematics is Internal Set Theory (IST). So far, we have not proved any theorems of internal mathematics by these new methods. Here is a ﬁrst example of that, a theorem of de Brujn and Erd˝os3 on the coloring of inﬁnite graphs. By a graph I mean a set G together with a subset R of G×G. Wedeﬁne a k-coloring of G, where k is a natural number, to be a function g:G → {1,...,k} such that g(x) 6= g(y) whenever hx,yi ∈ R. An example is G = R2 and R = {hx,yi : |x−y| = 1} where |x−y| is the Euclidean metric on R2. (This graph is known to have a 7-coloring but no 3-coloring.) The theorem asserts that if every ﬁnite subgraph of G has a k-coloring, so does G. This is not trivial, because if we color a ﬁnite subgraph, we may be forced to go back and change its coloring to color a larger ﬁnite subgraph containing it. To prove the theorem, we assume, by (T), that G and k are standard. By (I), there is a ﬁnite subgraph F of G containing all its standard elements, and by hypothesis F has a k-coloring f. Let g = Sf. Since f takes only standard values, every standard element of G is in the domain of g. By (T), every element of G is in the domain of g. To verify that g is a k-coloring, it suﬃces, by (T), to examine the standard elements, where it agrees with f. This concludes the proof.

 Exercises for Section 1.4 1. Show that if f is a function, so is Sf. What can one say about its domain and range? 2. Deduce (S) from (eS). 3. Deﬁne f by f(x) = t/π(t2 + x2) where t > 0 is inﬁnitesimal. What isR ∞ −∞ f(x)dx? What is Sf? 4. Establish external induction: Let A(n) be any formula, external or internal, containing the free variable n and possibly other parameters. Suppose that A holds for 0, and that whenever it holds for a standard n it also holds for n + 1. Then A(n) holds for all standard n. 5. Let the Greek variables range over ordinals. Establish external transﬁnite induction: ∃stαA(α) →∃stβ[A(β) & ∀stγ[A(γ) → β ≤ γ]]. 6. Deduce (I) from the forward direction of (I). 3N. G. de Brujn and P. Erd˝os, “A colour problem for inﬁnite graphs and a problem in the theory of relations,” Indagationes Mathematicae 13 (1951), 371–373.

 §1.5 ELEMENTARY TOPOLOGY 13 1.5 Elementary topology

 In this section I shall illustrate IST with some familiar material. In the following deﬁnitions by (S), we assume that E ⊆R, that f:E →R, and that x,y ∈ E. For f and x standard, f is continuous at x in case whenever y ' x we have f(y)' f(x). For f and E standard, f is continuous on E in case for all standard x, whenever y ' x we have f(y)' f(x). For f and E standard, f is uniformly continuous on E in case whenever y ' x we have f(y)' f(x). For E standard, E is compact in case for all x in E there is a standard x0 in E with x ' x0. For E standard, E is open in case for all standard x0 in E and all z in R, if z ' x0 then z is in E. For E standard, E is closed in case for all x in E and standard x0 in R, if x ' x0 then x0 is in E. For E standard, E is bounded in case each of its elements is limited. It can be shown that these deﬁnitions are equivalent to the usual deﬁnitions, but rather than worry about that now, let us develop a direct intuition for these formulations by examining them in various familiar situations. Let us prove that [0,1] is compact. Notice that it is standard. Let x be in [0,1]. Then it can be written as x =P∞ n=1 an2−n where each an is 0 or 1, and conversely each number of this form is in [0,1]. This binary expansion is determined by a function a:N+ →{0,1}. Let b = Sa. Then bn = an for all standard n, so if we let x0 =P∞ n=1 bn2−n then x0 is standard, x0 ' x, and x0 is in [0,1]. Therefore [0,1] is compact. This has the following corollary. Theorem 5. Each limited real number is inﬁnitely close to a unique standard real number. Proof. Let x be limited. Then [x] is a limited, and therefore standard, integer. Since x−[x] is in the standard compact set [0,1], there is a standard y0 inﬁnitely close to it, so if we let x0 = y0 +[x] then x0 is standard and inﬁnitely close to x. The uniqueness is clear, since 0 is the only standard inﬁnitesimal.

 14 CHAPTER 1. INTERNAL SET THEORY

 If x is limited, the standard number that is inﬁnitely close to it is called the standard part of x, and is denoted by stx. Now let us prove that a continuous function f on a compact set E is bounded. By (T), we assume them to be standard. Let K '∞, and let x be in E. It suﬃces to show that f(x) ≤ K. There is a standard x0 in E with x ' x0, and since f(x0) is standard we have f(x0) K. But by the continuity of the standard function f we have f(x)' f(x0), so f(x) ≤ K, which concludes the proof. (It may seem like cheating to produce an unlimited bound. By (T), if a standard function f is bounded, then it has a standard bound. But this is a distinction that can be made only in nonstandard analysis.) Somewhat more ambitiously, let us show that a continuous function f on a compact set E achieves its maximum. Again, we assume them to be standard. By (I), there is a ﬁnite subset F of E that contains all the standard points, and the restriction of f to the ﬁnite set F certainly achieves its maximum on F at some point x. By compactness andcontinuity, thereisastandard x0 in E with x0 ' x and f(x0)' f(x). Therefore f(x0) > ∼ f(y) for all y in F. Since every standard y is in F, wehave f(x0) > ∼ f(y) whenever y is standard, but since both numbers are standard we must have f(x0) ≥ f(y) whenever y is standard. By (T), this holds for all y, and the proof is complete. The same device can be used to prove that if f is continuous on [0,1] with f(0)≤0 and f(1)≥0, then f(x) = 0 for some x in [0,1]. The proof that a continuous function on a compact set is uniformly continuous requires no thought; it is a simple veriﬁcation from the definitions. It is equally easy to prove that a subset of R is compact if and only if it is closed and bounded (use Theorem 5 for the backward direction). All of this extends to Rn. Notice that Rn is standard if and only if n is, so in the deﬁnitions by (S), include “for n standard”. For a general metric space hX,di, where d is the metric, we deﬁne x ' y to mean that d(x,y) is inﬁnitesimal. Much of what was done above extends to metric spaces. In deﬁnitions by (S), include “for X and d standard”. For X and d standard, a metric space hX,di is complete in case for all x, if for all standard ε > 0 there is a standard y with d(x,y)≤ ε, then there is a standard x0 with x ' x0. This is equivalent to the usual deﬁnition. I shall sketch a proof of the Baire category theorem to illustrate an important point about nonstandard analysis. Let hX,di be a complete non-empty metric space, and let the Un be a sequence of open dense sets with intersection U. We want to show that U is non-empty. We denote the ε-neighborhood of x by N(ε,x). There are x1 and 0 < ε1 < 1/2

 §1.5 ELEMENTARY TOPOLOGY 15 such that N(ε1,x1)⊂ U1. Since U2 is open and dense, there are x2 and 0 < ε2 < 1/22 such that N(ε2,x2)⊂ U2 ∩N(ε1/2,x1). Continue in this way by induction, constructing a Cauchy sequence xn that converges to some point x, since X is complete. This x is in each Un, and so is in U. This proof is just the usual proof, and that is the important point. Nonstandard analysis is not an alternative to internal mathematics, it is an addition. By the nature of this book, almost all of the material consists of nonstandard analysis, but I emphatically want to avoid giving the impression that it should somehow be separated from internal mathematics. Nonstandard analysis supplements, but does not replace, internal mathematics. Let X be a standard topological space, and let x be a standard point in it. Then we deﬁne the relation y ' x to mean that y is in every standard neighborhood of x. The external discussion of continuity and compactness given above for R extends to this setting. Let E be a subset of the standard topological space X. Then we deﬁne the shadow of E, denoted by ◦E, as follows: ◦E = S{x ∈ X : y ' x for some y in E}. (1.10) Theorem 6. Let E be a subset of the standard topological space X. Then the shadow of E is closed. Proof. Let z be a standard point of X in the closure of ◦E. Then every open neighborhood of z contains a point of ◦E, so by (T) every standard open neighborhood U of z contains a standard point x of ◦E. But for a standard point x of ◦E, there is a y in E with y ' x, so y is in U. That is to say, ∀stU∃y[y ∈ E ∩U]. The open neighborhoods of z are a directed set under inclusion and this formula ﬁlters in U, so by the simpliﬁed version (1.4) of (I) we have ∃y∀stU[y ∈ E ∩U]; that is, ∃y∈E[y ' z]. Thus z is in ◦E. We have shown that every standard point z in the closure of ◦E is in ◦E, so by (T) every point z in the closure of ◦E is in ◦E. Thus ◦E is closed. There is a beautiful nonstandard proof of the Tychonov theorem. Let T be a set, let Xt for each t in T be a compact topological space, and let Ω be the Cartesian product Ω =Qt∈T Xt with the product topology. We want to show that Ω is compact. By (T), we assume that t 7→ Xt is standard, so that Ω is also standard. Let ω be in Ω. For all standard t there is a standard point y in Xt such that y ' ω(t), so by (eS) there is a standard η in Ω such that for all standard t we have η(t) ' ω(t). By the deﬁnition of the product topology, η ' ω, so Ω is compact.

 16 CHAPTER 1. INTERNAL SET THEORY

 Exercises for Section 1.5 1. Precisely how is (eS) used in the proof of the Tychonov theorem? 2. Let X be a compact topological space, f a continuous mapping of X onto Y . Show that Y is compact. 3. Is the shadow of a connected set connected? 4. Let x be in R. What is the shadow of {x}? 5. Let f(x) = t/π(t2 + x2), where t > 0 is inﬁnitesimal. What is the shadow of the graph of f? 6. Let E be a regular polygon of n sides inscribed in the unit circle. What is the shadow of E?

 1.6 Reduction of external formulas

 It turns out that (with a proviso which I shall discuss shortly) every external formula can be reduced to a weakly equivalent internal formula. This is accomplished by a kind of formal, almost algebraic, manipulation of formulas, using (eS) and (I) to push the external quantiﬁers ∀st and ∃st to the left of the internal quantiﬁers ∀ and ∃, and then using (T) to get rid of them entirely. In this way we can show that the deﬁnitions made using (S) in the previous section are equivalent to the usual ones. More interestingly, we can reduce the rather weird external theorems that we have proved to equivalent internal form. The proviso, which has only nuisance value, is that whenever we use (eS) to introduce a standard function ˜ y(x), then x and y must be restricted to range over a standard set, to give the function ˜ y a domain. (Actually, it suﬃces to make this restriction on x alone.) I will not make this explicit all of the time. To reduce a formula, ﬁrst eliminate all external predicates, replacing them by their deﬁnitions until only “standard” is left. Even this should be eliminated, replacing “x is standard” by ∃sty[y = x]. Second, look for an internal quantiﬁer that has some external quantiﬁers in its scope. (If this never happens, we are ready to apply (T) to obtain a weakly equivalent internal formula.) If the internal quantiﬁer is ∀, use (eS) to pull the ∀st’s to the left of the ∃st’s (if necessary, thereby introducing standard functions), where they can then be pulled to the left of ∀. If the internal quantiﬁer is ∃, proceed by duality. Third, whenever ∃y∀stxA (or its dual) occurs with A internal, use (I), taking advantage of its simpliﬁcation (1.4) whenever A ﬁlters in x. One thing to remember in using this reduction algorithm is that the implication ∃stxA(x) → B is equivalent to ∀stx[A(x) → B], and dually ∀stxA(x) → B is equivalent to ∃stx[A(x) → B], provided in both cases that x does not occur free in B. (This has nothing to do with the superscript “st”; it is a general fact about quantiﬁers in the hypothesis of an

 §1.6 REDUCTION OF EXTERNAL FORMULAS 17 implication. For example, let A(s) be “s is an odd perfect number” and let B be the Riemann hypothesis. Then ∃sA(s)→B and ∀s[A(s)→B] say the same thing.) If B begins with an external quantiﬁer, it comes out of the implication unchanged, but we have our choice of which quantiﬁer to take out ﬁrst. The idea is to do this in such a way as to introduce as few functions as possible. If we have an equivalence, we must rewrite it as the conjunction of two implications and rename the bound variables, since they come out of the implications in diﬀerent ways. For example, the formula ∀stxA(x)↔∀styB(y) is equivalent to [∀stxA(x)→∀styB(y)] & [∀stuB(u)→∀stvA(v)], which in turn is equivalent to ∃stx∃stu∀sty∀stv[A(x)→B(y)] & [B(u)→A(v)) and also to ∀sty∀stv∃stx∃stu[A(x)→B(y)] & [B(u)→A(v)) . If all of this is preceded by ∀t, where t is a free variable in A or B, the second form is advantageous; if it is preceded by ∃t, one should use the ﬁrst form. Let us illustrate the reduction algorithm with the deﬁnitions by (S) in the previous section of “continuous at x”, “continuous”, and “uniformly continuous”. The formulas in question are respectively ∀yy ' x → f(y)' f(x) ,∀ stx∀yy ' x → f(y)' f(x) , ∀x∀yy ' x → f(y)' f(x) .We must eliminate '. With ε and δ ranging over R+, we obtain ∀y∀stδ[|y−x|≤ δ]→∀stε[|f(y)−f(x)|≤ ε] , (1.11)∀ stx∀y∀stδ[|y−x|≤ δ]→∀stε[|f(y)−f(x)|≤ ε] , (1.12) ∀x∀y∀stδ[|y−x|≤ δ]→∀stε[|f(y)−f(x)|≤ ε] . (1.13) In all of these formulas, f is a standard parameter, and x is a standard parameter in (1.11). First bring ∀stε out; it goes all the way to the left.

 18 CHAPTER 1. INTERNAL SET THEORY Then bring∀stδ out; it changes to∃stδ, and since the formula ﬁlters in δ, it goes to the left of ∀y, and also of ∀x in (1.13). This yields ∀stε∃stδ∀y|y−x|≤ δ →|f(y)−f(x)|≤ ε ,∀ stε∀stx∃stδ∀y|y−x|≤ δ →|f(y)−f(x)|≤ ε , ∀stε∃stδ∀x∀y|y−x|≤ δ →|f(y)−f(x)|≤ ε . Now apply transfer; this simply removes the superscripts “st” and shows that our deﬁnitions are equivalent to the usual ones. Now consider ∀x∃stx0[x ' x0]; that is, ∀x∃stx0∀stε|x−x0|≤ ε . (1.14)It is understood that x and x0 range over the standard set E; this was our deﬁnition by (S) of E being compact. First we move ∀stε to the left, by (eS), introducing the standard function ˜ ε:E → R+. Thus (1.14)becomes ∀st˜ ε∀x∃stx0|x−x0|≤ ˜ ε(x0) . (1.15)Now use (I). There is no ﬁltering in (1.15), so it becomes ∀st˜ ε∃stﬁnx0 0∀x∃x0∈x0 0|x−x0|≤ ˜ ε(x0) .Now apply (T). We obtain ∀˜ ε∃ﬁnx0 0∀x∃x0∈x0 0|x−x0|≤ ˜ ε(x0) ,where ∃ﬁn means “there exists a ﬁnite set such that” (and similarly ∀ﬁn means “for all ﬁnite sets”). But this is mathematically equivalent to the usual deﬁnition of a set being compact. Deﬁnitions by (S) are an external way of characterizing internal notions, but they simultaneously suggest new external notions. These are often indicated by the preﬁx S-. Thus we say that f is S-continuous at x in case whenever y ' x we have f(y) ' f(x). If both f and x are standard, this is the same as saying that f is continuous at x. But let t > 0 be inﬁnitesimal, and let f(x) = t/π(t2 + x2). Then f is continuous at 0 (this internal property is true for any t > 0), but it is not S-continuous at 0. Let g(x) = t for x 6= 0 and g(0) = 0. Then g is discontinuous at 0 but is S-continuous at 0. Let h(x) = x2. Then h is continuous at 1/t but is not S-continuous at 1/t. Similarly, we say that f is S-uniformly continuous in case whenever y ' x we have f(y)' f(x).

 §1.6 REDUCTION OF EXTERNAL FORMULAS 19 I am not very fond of the S-notation. It seems to imply that to every internal notion there is a unique corresponding external S-notion, but this is not so. There may be distinct external notions that are equivalent for standard values of the parameters. Now let us apply the reduction algorithm to some of our external theorems. Let us split Theorem 2 into three diﬀerent statements. (i) If every element of a set is standard, then the set is ﬁnite. That is, ∀S∀y∈S∃stx[y = x]→ S is ﬁnite , ∀S∃y∈S∀stxy = x → S is ﬁnite ,∀ stﬁnx0∀S∃y∈S∀x∈x0y = x → S is ﬁnite , (1.16) ∀ﬁnx0∀S∃y∈S∀x∈x0y = x → S is ﬁnite , (1.17) ∀ﬁnx0∀SS ⊆ x0 → S is ﬁnite .In other words, (i) is equivalent to (i 0): every subset of a ﬁnite set is ﬁnite. We used (I) for (1.16), (T) for (1.17), and then pushed quantiﬁers back inside the implication as far as possible to make the result more readable. (ii) If every element of a set is standard, then the set is standard. That is, ∀S∀y∈S∃stx[y = x]→∃stT[S = T] . (1.18)Apply (I) to the hypothesis: ∀S∃stﬁnx0∀y∈S∃x∈x0[y = x]→∃stT[S = T] ;simplify: ∀S∃stﬁnx0[S ⊆ x0]→∃stT[S = T] ; pull out the quantiﬁers and apply (I): ∀stﬁnx0∃stﬁnT0∀S∃T∈T0[S ⊆ x0 → S = T]; simplify and apply (T): ∀ﬁnx0∃ﬁnT0∀S[S ⊆ x0 → S ∈ T0]. This is true for T0 = ℘x0, where ℘X denotes the power set of X (the set of all subsets of X). In other words, (ii) is equivalent to (ii0): the power set of a ﬁnite set is ﬁnite. When reducing an external formula, it is a

 20 CHAPTER 1. INTERNAL SET THEORY

 good idea to attack subformulas ﬁrst. For example, had we pulled the quantiﬁers out of (1.18) directly, we would have obtained ∀S∃stT∃y∈S∀stx[y = x → S = T], which after reduction and simpliﬁcation gives: ∀e x0∃ﬁnT0∀ShS ⊆ \ T∈T0e x0(T)→ S ∈ T0i. (1.19) Every function must have a domain, so the use of (eS) to produce the ﬁnite-set valued function e x0 was illegitimate; to be honest, we should have introduced ∀stV with all of the variables restricted to lie in V , so that after transfer the formula begins ∀V with all of the variables restricted to lie in V . Consider this done. Here is an internal proof of (1.19). Choose any T0 and let T0 = ℘e x0(T0)∪{T0}. If S ⊆TT∈T0e x0(T), then in particular S ⊆e x0(T0) and so S ∈ T0. Thus (1.19) is an obscure way of saying that the power set of a ﬁnite set is ﬁnite. (iii) Every element of a standard ﬁnite set is standard. That is, ∀stSS is ﬁnite→∀y∈S∃stx[y = x] .This reduces easily to ∀S∃ﬁnx0[S is ﬁnite→ S ⊆ x0]. In other words, (iii) is equivalent to (iii0): every ﬁnite set is a subset of some ﬁnite set. With the variables ranging over N, Theorem 3 is ∀n∀kk ≤ n →∃stj[j = k] ,which reduces easily to ∀n∃ﬁnj0∀k[k ≤ n → k ∈ j0]. Theorem 4, that there is a ﬁnite set containing every standard object, reduces immediately to the following triviality: for all ﬁnite sets x0 there is a ﬁnite set F such that every element of x0 is an element of F.

 §1.6 REDUCTION OF EXTERNAL FORMULAS 21 Theorem 4 is shocking, but the informal statement “there is a ﬁnite set containing any ﬁxed element” appears quite reasonable. Similarly, with the variables ranging over R+, one is accustomed to saying “there is an x less than any ﬁxed ε.” Nonstandard analysis takes the further step of saying “call such an x an inﬁnitesimal.” I want to emphasize again that the predicate “standard” has no semantic content in IST; it is a kind of syntactical place-holder signifying that the object in question is to be held ﬁxed. With many objects in play at once, some depending on others, the syntax of being held ﬁxed becomes complicated, and the rules for handling the idea correctly are (I), (S), and (T). What Abraham Robinsoninventedisnothinglessthananewlogic. Hewasexplicitabout this in the last paragraph of his epoch-making book:4 Returning now to the theory of this book, we observe that it is presented, naturally, within the framework of contemporary Mathematics, and thus appears to aﬃrm the existence of all sorts of inﬁnitary entities. However, from a formalist point of view we may look at our theory syntactically and may consider that what we have done is to introduce new deductive procedures rather than new mathematical entities. One technical point is worth commenting on. It is essential to the success ofthereductionalgorithmthattheidealizationprincipleholdforinternal formulas with free variables; this feature was not present in Robinson’s notion of enlargement [Ro, §2.9]. Exercises for Section 1.6 1. Find a function f that is S-uniformly continuous without being uniformly continuous, and vice versa. 2. What is the reduction of the formula x ' 0? 3. What is the reduction of Theorem 5? 4. Show that the deﬁnitions by (S) of open, closed, and bounded in the previous section are equivalent to the usual ones. 5. Show that the deﬁnition by (S) of a complete metric space in the previous section is equivalent to the usual one. 6. Say that a sequence a:N → R is S-Cauchy in case for all unlimited n and m we have an ' am. Show that a standard sequence is S-Cauchy if and only if it is Cauchy. Say that a is of limited ﬂuctuation in case for all standard ε > 0 and all k, if n1 < ••• < nk and |an1 −an2|≥ ε, ..., |ank−1 −ank|≥ ε, then k is limited. Show that a standard sequence is of limited ﬂuctuation if and only if it is Cauchy. Can a sequence be S-Cauchy without being of limited ﬂuctuation, or vice versa? 7. Is the unit ball of Rn, where n is unlimited, compact? 4[Ro] Abraham Robinson, ”Non-Standard Analysis,” Revised Edition, American Elsevier, New York, 1974.

 22 CHAPTER 1. INTERNAL SET THEORY

 8? Can the reduction of Theorem 6 be made intelligible? (The ? means that I don’t know the answer.)

 1.7 Answers to the exercises

 Section 1.1 1. Yes and yes; this set can be formed for any natural number, and it is a ﬁnite set. 2. No. 3. Let x be standard and positive. Then x ≤ x, so x is limited. 4. No. 5. At ﬁrst sight this looks like illegal set formation, but let S be the set of all real numbers x such that x2 ≤ 1. Assuming that 1 is standard, we see that every x in S is limited, so S is the set in question. 6. Not yet; so far, nothing guarantees that if ε is standard then so is ε/2.

 Section 1.2 1. Yes. Let x and y be inﬁnitesimal and let ε > 0 be standard. By transfer, ε/2 is standard, so that |x|≤ ε/2 and |y|≤ ε/2. Then |x + y|≤ ε, and since ε was an arbitrary strictly positive standard number, x + y is inﬁnitesimal. 2. There are standard numbers R and S such that |r|≤ R and |s|≤ S. By (T), R + S and RS are standard. 3. Let ε > 0 be standard. There is a standard R, which we may take to be non-zero, such that |r|≤ R. By (T), ε/R is standard, so |x|≤ ε/R. Hence |xr|≤ ε. 4. For ε > 0 and r = 1/ε, ε is standard if and only if r is standard, by the transfer principle. 5. We cannot speak of “the inﬁnitesimals” as an ideal, or of “the integral domain of limited real numbers”, without illegal set formation. If we avoid these illegal set formations, the preceding exercises essentially give an aﬃrmative answer to the ﬁrst question. But to talk about “the quotient ﬁeld” we really need sets. The discussion must be postponed to Chapter 4, where we shall have external sets at our disposal. 6. No. If the closed interval is [0,b] where b is unlimited, the function has no standard bound. The transfer was illegal because the closed interval was a parameter. 7. We have x ≤ y + α where α is inﬁnitesimal and x and y are standard. Then x ≤ y + ε for all standard ε > 0, so by (T) we have x ≤ y + ε for all ε > 0. Hence x ≤ y.

 §1.7 ANSWERS TO THE EXERCISES 23

 Section 1.3 1. This is just a restatement of Theorem 3. 2. Yes. Euclid showed that for any natural number n there is a prime greater than n. 3. Let x > 0 be inﬁnitesimal. Then 0 < x < 1, so x is of the form x =P∞ n=1 an10−n where each an is one of 0, ..., 9. Since 10−n is standard if n is, by (T), each an with n standard is 0. But x > 0, so not all of its decimal digits are 0. As we already know, any x > 0 has a ﬁrst non-zero decimal digit. What about the number whose decimal digits are 0 for all standard n and 7 for all nonstandard n? The question makes so sense. The decimal digits form a sequence; a sequence is a set; and I committed illegal set formation in posing the question. 4. Yes; by (I), there is a ﬁnite subset of H containing all of its standard points, so take its span. (This is by no means unique, and we cannot form the smallest such space without illegal set formation.) Any ﬁnite dimensional subspace of a Hilbert space is closed. 5. We have ∀x∃stn[x ∈ Sn], where x ranges over S. By (I), we have that ∃stﬁnn0∀x∃n∈n0[x ∈ Sn], and since the sets are disjoint, all but a standard ﬁnite number of them are empty. 6. The set of all N such that |an| ≤ 1/n for all n ≤ N contains all standard N, so by overspill it contains some unlimited N. 7. Not necessarily; consider the identity function. 8. We can assume that X is non-empty, since otherwise both sides are vacuously true. The backward direction holds by (I). In the forward direction, by (I) we have some ﬁnite standard x0 that works, and we can take it to be non-empty. But by Theorem 2, x0 ∩X is also a standard ﬁnite set, and it is an element of ℘ﬁnX.

 Section 1.4 1. Let f:X → Y , and assume that X and Y are contained in some standard set. Let g = Sf. For all z, if z ∈ g then z ∈ SX × SY ; for all x in SX and y1 and y2 in SY , if hx,y1i and hx,y2i are in g, then y1 = y2. These statements hold by deﬁnition for the standard elements, and since the sets in question are standard, they hold by (T) for all elements. Thus Sf is a function from a subset of SX into SY . 2. Replace Y by its characteristic function. Let X be a standard set, let z range over X, let y range over {0,1}, and let ˜ y range over {0,1}X. We clearly have ∀stz∃sty[y = 1 ↔ A], so by (eS) we have ∃st˜ y∀stz[˜ y(z) = 1 ↔ A]. Then we let Y = {z ∈ X : ˜ y(z) = 1}. 3. By elementary calculus, this integral is equal to 1 for any t > 0. There are no standard pairs hx,f(x)i, so Sf is the empty set. 4. Let S = S{n ∈ N : A(n)}. Then 0 is in S. By assumption, for all standard n, if n is in S then n + 1 is in S, so by (T) this is true for all n. By induction, S = N. In particular, every standard n is in S, so A(n) holds for every standard n.

 24 CHAPTER 1. INTERNAL SET THEORY

 5. Let

 S = S{γ ≤ α : A(γ)}. Then every standard element of S is an ordinal, so the same is true by (T) for all elements of S. Since S is non-empty (it contains α), it contains a least β, which is standard by (T). Then β ≤ α. Consider a standard γ such that A(γ). If α < γ, then certainly β ≤ γ; if γ ≤ α then γ ∈ S, so again β ≤ γ. This concludes the proof. This is often used to prove ∀stαB(α). Argue indirectly. If not, there is a least standard α such that A(α) where A is ¬B. If we can show that α cannot be 0, a successor, or a limit ordinal, then the proof will be complete. 6. To establish (I), we need only show (without using the backward direction of the idealization principle) that every element of a standard ﬁnite set x0 is standard. We do this by external induction on the cardinality n of x0. By transfer, n is standard. The statement is vacuously true for n = 0. For n > 0, the set x0 contains an element, so by (T) it contains a standard element. Delete this element; what remains is a set of cardinality n−1 and by (T) it is a standard set. By the external induction hypothesis, all of the remaining elements are also standard.

 Section 1.5 1. An element η of Ω is a function from T toSt∈T Xt such that η(t) ∈ Xtfor all t in T. So if we know that for all standard t there is a y in Xt with y ' ω(t), then (eS) tells us that there is a standard η (a ˜ y) in Ω such that η(t) ' ω(t) for all standard t. By the deﬁnition of the product topology and (T), a standard basic neighborhood of η is given by a standard ﬁnite set of ti in T and a standard ﬁnite set of neighborhoods Ui of the η(ti), so η ' ω. 2. By (T), assume that f, X, and Y are standard. Let y be in Y . Then there is an x in X with f(x) = y and a standard x0 in X with x0 ' x. By (T), f(x0) is standard, and by continuity, f(x0) ' f(x). Hence Y is compact. 3. Not necessarily. In the plane, consider the x-axis and the parallel line one unit above it together with the vertical interval of length 1 with x unlimited joining them. 4. If x is limited, then ◦{x} is {stx}; otherwise it is the empty set. 5. If we graph f, what we see is the union of the x-axis and the positive half of the y-axis, and this is the shadow of the graph of f. Let x 6= 0 be standard; then hx,f(x)i ' hx,0i. Let y > 0 be standard; then the positive solution x of f(x) = y is inﬁnitesimal, so hx,f(x)i ' h0,yi. Theorem 6 tells us that the origin too must be in the shadow. To see this directly, choose x = t1/4, for example; then hx,f(x)i'h0,0i. 6. Suppose that n is standard. If one, and hence all, of the vertices is standard, then E is standard and since it is closed ◦E = E; otherwise ◦E is obtained by an inﬁnitesimal rotation of E. If n is nonstandard, ◦E is the unit circle.

 §1.7 ANSWERS TO THE EXERCISES 25

 Section 1.6 1. Let t > 0 be inﬁnitesimal, and let f(x) = t/π(t2 + x2). Then f is uniformly continuous but not S-uniformly continuous. For an example in the other direction, let g(x) = t for x 6= 0 and g(0) = 0. 2. The formula x ' 0, i.e., ∀stε[|x|≤ ε] where ε ranges over R+, is weakly equivalent to the internal formula ∀ε[|x| ≤ ε], which is equivalent to x = 0. The only standard inﬁnitesimal is 0. 3. We have ∀x[∃str[|x|≤ r] →∃stx0∀stε[|x−x0|≤ ε]], ∀str∀x∃stx0∀stε[|x|≤ r →|x−x0|≤ ε], ∀str∀st˜ ε∃stﬁnx0 0∀x∃x∈x0 0[|x|≤ r →|x−x0|≤˜ ε(x0)]; then remove the superscripts “st”. This says that every interval [−r,r] is compact. 4. For “open” we have ∀stx0∈E∀z∈R[∀stε[|z−x0|≤ ε] → z ∈ E]. Bring ∀stε out as ∃stε, which ﬁlters past ∀z∈R. Then use (T) to obtain ∀x0∃ε∀z∈R[|z−x0|≤ ε → z ∈ E].

 Similarly, the formula for “closed” reduces to ∀x0∈R∃ε∀x∈E[|x−x0|≤ ε → x0 ∈ E], which is perhaps more readable if we push ∀x∈E inside the parentheses as ∃x∈E. For “bounded”, use the ﬁltering form of (I) and (T) to obtain the usual formulation ∃r∀x∈E[|x|≤ r]. 5. Let hX,di be a standard metric space. To avoid confusion, let us temporarily call it nice in case for all x, if for all standard ε > 0 there is a standard y with d(x,y) ≤ ε, then there is a standard x0 with x ' x0. Suppose that X is incomplete. By (T), there is a standard Cauchy sequence xn with no limit. Let ν ' ∞ and let ε > 0 be standard. Consider the set S of all n such that d(xn,xν) ≤ ε. This set contains all unlimited n, so by overspill it contains some limited n, for which xn is standard. But there is no standard x0 with xν ' x0, for then it would be the limit of the xn. Thus an incomplete standard metric space is not nice. Conversely, let X be complete and let x be a point in X such that for all standard ε > 0 there is a standard y with d(x,y) ≤ ε. Then for all standard n there is a standard y with d(x,y) ≤ 1/n, so by (eS) there is a standard sequence yn such that d(x,yn) ≤ 1/n for all standard n. By the triangle inequality and (T), yn is a Cauchy sequence and so has a limit x0, which is standard by (T), and x ' x0. Thus a complete

 26 CHAPTER 1. INTERNAL SET THEORY

 standard metric space is nice, and our deﬁnition by (S) of a metric space being complete is equivalent to the usual one. It is also interesting to approach this problem via the reduction algorithm. We have ∀x[∀stε∃sty[d(x,y) ≤ ε]→∃stx0∀stδ[d(x,x0) ≤ δ]]. Use (eS) before pulling anything out, to avoid spurious arguments of the functions, and then pull out the resulting functions. We obtain ∀st˜ δ∀st˜ y∀x∃stx0∃stε[d(x,˜ y(ε)) ≤ ε → d(x,x0) ≤ ˜ δ(x0)],

 which by (I) and (T) is equivalent to ∀˜ δ∀˜ y∃ﬁnε0∃ﬁnx0 0∀x∃x0∈x0 0[∀ε∈ε0(d(x,˜ y(ε)) ≤ ε)→ d(x,x0) ≤ ˜ δ(x0)], where ∃ε∈ε0 was pushed back inside the implication. Now we have the internal problem of seeing that this is equivalent to completeness of the metric space. Only the Cauchy ˜ y are relevant (to obtain a more customary notation, we could let yn = ˜ y(1/n)), for otherwise we can ﬁnd a two-point set ε0 that violates the hypothesis of the implication, by the triangle inequality. If the space X is complete, just let x0 0 be the singleton consisting of the limit as ε → 0 of ˜ y(ε). To construct a counterexample when X is not complete, let ˜ y be Cauchy without a limit, and let ˜ δ(x0) be the distance from x0 to the limit (in the completion) of ˜ y(ε). 6. The S-Cauchy condition is ∀n∀m[∀str[n ≥ r & m ≥ r] →∀stε[|an −am|≤ ε]].

 The r ﬁlters out to give the usual deﬁnition of a Cauchy sequence. The limited ﬂuctuation condition is ∀stε∀k[A(k,ε,a) →∃str[k ≤ r]],

 where A(k,ε,a) is an abbreviation for the assertion that the sequence a contains k ε-ﬂuctuations. Again the r ﬁlters out, and a standard sequence is of limited ﬂuctuation if and only if for all ε > 0 there is a bound r on the number of ε-ﬂuctuations, which is the same as being Cauchy. Now let a be an S-Cauchy sequence, not necessarily standard, and let ε > 0 be standard. Consider the set of all n such that for all m > n we have |an −am| ≤ ε; this set contains all unlimited n, so by overspill it contains some limited n. Consequently, an S-Cauchy sequence is of limited ﬂuctuation. But let ν ' ∞ and let an = 0 for n < ν and an = 1 for n ≥ ν. This sequence is of limited ﬂuctuation but is not S-Cauchy. Thus we have two distinct external concepts that agree on standard objects. 7378.

Nelson E., “Internal set theory. A new approach to nonstandard analysis,” Bull. Amer. Math. Soc., 83, No. 6, 1165–1198 (1977).

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