An Invitation to Nonstandard Analysis and its Recent Applications

Isaac Goldbring and Sean Walsh

Introduction approximate groups. However, its roots go back to Robin- Nonstandard analysis has been used recently in major re- son’s formalization of the approach to calcu- sults, such as Jin’s sumset theorem in additive combina- lus. After first illustrating its very basic uses in torics and Breuillard–Green–Tao’s work on the structure of in “Calculus with ,” we go on to highlight a selection of its more serious achievements in “Selected Classical and Recent Applications,” including the afore- Isaac Goldbring is an associate professor in the Department of Mathematics at mentioned work of Jin and Breuillard–Green–Tao. After UC Irvine. His email address is [email protected]. presenting a simple axiomatic approach to nonstandard Sean Walsh is an associate professor in the Department of Philosophy at UCLA. analysis in “ for Nonstandard Extensions” we ex- His email address is [email protected]. amine Jin’s theorem in more detail in “The Axioms in Ac- For permission to reprint this article, please contact: tion: Jin’s Theorem.” Finally, in “The Con- [email protected]. struction” we discuss how these axioms can be justified DOI: https://doi.org/10.1090/noti1895

842 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, 6 with a particular concrete construction (akin to the verifica- To illustrate this method, let us say that two elements tion of the axioms for the real using Dedekind cuts or 푎 and 푏 of 퐑∗ are infinitely close to one another, denoted Cauchy sequences), and in “Other Approaches” we com- 푎 ≈ 푏, if their difference is an infinitesimal. Then one has: pare our axiomatic approach to other approaches. While Theorem 1. Suppose that 푓 ∶ 퐑 → 퐑 is a function and 푎 ∈ brief, our hope is that this survey can quickly give the read- 퐑. Then 푓 is continuous at 푎 if and only if: whenever 푏 ∈ 퐑∗ er a sense of both how nonstandard methods are being and 푎 ≈ 푏, then 푓(푎) ≈ 푓(푏). Likewise, 푓 is differentiable used today and how these methods can be rigorously pre- at 푎 with 퐿 if and only if: whenever 푏 is a nonzero sented and justified. ∗ 푓(푎+푏)−푓(푎) infinitesimal in 퐑 , one has that 푏 ≈ 퐿. Calculus with Infinitesimals Proof. We only discuss the continuity statement, since the Every mathematician is familiar with the fact that the differentiability statement is entirely analogous. Suppose founders of calculus such as Newton and Leibniz made that the assumption of the “if” direction holds and fix 휖 > free use of infinitesimal and infinite quantities, e.g., in ex- 0 in 퐑. In 퐑∗, by choosing an infinitesimal 훿 > 0, the ′ pressing the derivative 푓 (푎) of a differentiable function following elementary statement about 휖 is true: “there is 푓 at a point 푎 as being the that is infinitely 훿 > 0 such that, for all 푏, if |푎 − 푏| < 훿, then |푓(푎) − 푓(푎+훿)−푓(푎) close to 훿 for every nonzero infinitesimal “quan- 푓(푏)| < 휖.” The logical similarity mentioned above then tity” 훿. Here, when we say that 훿 is infinitesimal, we mean implies that this elementary statement about 휖 is also true that |훿| is less than 푟 for every positive real number 푟. in 퐑. But this is precisely what is needed to prove that 푓 is Of course, the mathematical status of such quantities continuous at 푎. The other direction is similar. □ was viewed as suspect and the entirety of calculus was put ∗ on firm foundations in the nineteenth century by the likes Every finite element 푎 of ℝ is infinitely close to a unique of Cauchy and Weierstrass, to name a few of the more real number, called its standard part, denoted st(푎). Thus, significant figures in this well-studied part of the history the first part of the above theorem reads: 푓 is continuous of mathematics. The innovations of their “휖-훿 method” at 푎 if and only if: whenever st(푏) = 푎, then st(푓(푏)) = (much to the chagrin of many real analysis students today) 푓(푎). In these circumstances, one can then write 푏 = 푎 + allowed one to give rigor to the naïve arguments of their 훿푥 and 푓(푏) = 푓(푎)+훿푦 where 훿푥, 훿푦 are infinitesimals. predecessors. In his elementary calculus textbook based on nonstandard In the 1960s, realized that the ideas methods, Keisler pictured this as a “microscope” by which and tools present in the area of logic known as model the- one can zoom in and study the local behavior of a function ory could be used to give precise mathematical meanings to at a point. infinitesimal quantities. Indeed, one of Robinson’s stated Besides extensionally characterizing familiar concepts aims was to rescue the vision of Newton and Leibniz ([20]). such as continuity and differentiability, the use of infinites- While there are complicated historical and philosophical imals can help to abbreviate proofs. To illustrate, let us questions about whether Robinson succeeded entirely in prove the following, noting how we can avoid the usual 휖 this (cf. [5]), our goal is to discuss some recent applica- “ 2 ” argument and instead appeal to the simple fact that tions of the methods Robinson invented. finite sums of infinitesimals are again infinitesimal: To this end, let us turn now to the basics of Robinson’s Corollary 2. Suppose that 푓 ∶ 퐑 → 퐑 and 푔 ∶ 퐑 → 퐑 are approach. On this approach, an infinitesimal 훿 is merely 푎 ∈ 퐑 푓 + 푔 ∗ both continuous at . Then is also continuous at an element of an ordered field 퐑 properly containing 푎. the ordered field 퐑 of real . While such nonar- chimedean fields were already present in, for example, al- Proof. Consider 푏 ≈ 푎. By assumption, we have 푓(푏) ≈ gebraic number theory, the new idea that allowed one to 푓(푎) and 푔(푏) ≈ 푔(푎). Then since the sums of two in- correctly apply the heuristics from the early days of calcu- finitesimals is infinitesimal, we have ∗ lus was that 퐑 is logically similar to 퐑, in that any ele- (푓+푔)(푏) = 푓(푏)+푔(푏) ≈ 푓(푎)+푔(푎) = (푓+푔)(푎). mentary statement true in the one is also true in the other. 푏 푓 + 푔 (Formally, elementary statements are defined using first- Since was arbitrary, we see that is continuous at 푎 □ order logic: see “Axioms for Nonstandard Extensions” for . a self-contained presentation.) A particular feature of this Selected Classical and Recent Applications approach is that one functorially associates to every func- The primary reason for the contemporary interest in non- tion 푓 ∶ 퐑 → 퐑 an extension 푓 ∶ 퐑∗ → 퐑∗ (and similarly standard analysis lies in its capacity for proving new re- for relations). In this light, 퐑 is referred to as the standard sults. In this section, we briefly describe a handful of the field of real numbers whilst 퐑∗ is referred to as a nonstandard more striking results that were first proven using nonstan- field of real numbers or a hyperreal field. dard methods. We remark that nonstandard methods have

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 843 proven useful in nearly every area of mathematics, includ- only the large-scale geometry of the space. The prime ex- ing algebra, measure theory, functional analysis, stochas- ample of this phenomenon is that an asymptotic cone of tic analysis, and mathematical economics to name a few. the discrete space 퐙 is the continuum 퐑. Here, we content ourselves with a small subset of these ap- In [8], van den Dries and Wilkie used the nonstandard plication areas. perspective to give a much cleaner account of the asymp- Bernstein-Robinson theorem on invariant subspaces. totic cone construction. Indeed, given a metric space (푋, 푑), The famous Invariant Subspace Problem asks whether every a fixed point 푥0 ∈ 푋, and an infinite ∗ bounded operator 푇 on a separable 퐻 has 푅 ∈ 퐑 , one can look at the subspace a 푇-invariant closed subspace besides {0} and 퐻. In the ∗ 푑(푥, 푥0) 1930s, von Neumann proved that compact operators have 푋푅 ∶= {푥 ∈ 푋 ∶ is finite}. nontrivial closed invariant subspaces. (Recall an operator 푅 is compact if it is the norm- of finite-rank operators.) One can then place the metric 푑푅 on 푋푅 given by 푑푅(푥, 푦) Little progress was made on the Invariant Subspace Prob- 푑(푥,푦) ∶= st( ). When 푋 is the Cayley graph associated to a lem until: 푅 group, then the polynomial growth condition on Γ allows 푅 푋 Theorem 3 (Bernstein-Robinson [3]). If 푇 is a polynomi- one to find so that 푅 is locally compact; the verifica- 푋 ally compact operator on 퐻 (meaning that there is a nonzero tion of this and other important properties of 푅 are very polynomial 푝(푍) ∈ ℂ[푍] such that 푝(푇) is a compact oper- clear from the nonstandard perspective. While Gromov’s ator), then 푇 has a nontrivial closed invariant subspace. original proof did not use nonstandard methods, the non- standard perspective on asymptotic cones has now in fact One of the main ideas of the proof is to use the nonstan- become the one that is presented in courses and textbooks dard version of the basic fact that operators on finite- on the subject. See, for instance, [18]. dimensional spaces have upper-triangular representations Jin’s Sumset Theorem. In additive combinatorics, the fo- to find many hyperfinite-dimensional subspaces of 퐻∗ cus is often on densities and structural properties of sub- which are 푇∗-invariant. (Upon seeing the Berenstein- sets of 퐍. Given 퐴 ⊆ 퐍, we define the Banach density of 퐴 |퐴∩[푥,푥+푛)| Robinson theorem, Halmos proceeded to give a proof that to be BD(퐴) ∶= lim푛→∞ max푥∈퐍 푛 . If BD(퐴) > did not pass through nonstandard methods and the two 0, then we think of 퐴 as a “large” subset of 퐍. An impor- papers were published together in the same volume.) We tant structural property of sets of natural numbers is that should point out that the Berenstein-Robinson theorem of being piecewise syndetic, where 퐴 is piecewise syndetic if was later subsumed by Lomonosov’s theorem from 1973, there is 푚 ∈ 퐍 such that 퐴 + [0, 푚] contains arbitrarily which says that an operator that commutes with a nonzero long intervals. Renling Jin [14] used nonstandard analysis compact operator satisfies the conclusion of the Invariant to prove the following: Subspace Problem. 퐴, 퐵 ⊆ 퐍 Asymptotic cones. In [11], Gromov proved the following Theorem 5. If both have positive Banach density, 퐴 + 퐵 theorem, which is one of the deepest and most beautiful then is piecewise syndetic. theorems in geometric group theory: Jin’s theorem has paved the way for further applications of Theorem 4. If 퐺 is a finitely generated group of polynomial nonstandard methods in additive combinatorics, which is growth, then 퐺 is virtually solvable. now a very active area; see the monograph [6]. In “The Axioms in Action: Jin’s Theorem,” we discuss the proof Here, a finitely generated group has polynomial growth if the of Jin’s theorem, using the formal framework for nonstan- set of group elements that can be written as a product of at dard analysis that we set out in “Axioms for Nonstandard most 푑 generators and their inverses grows polynomially Extensions.” in 푑. The polynomial growth condition is a geometric con- The structure of approximate groups. Fix 퐾 ≥ 1. A sym- dition describing the growth of the sizes of balls centered metric subset 퐴 of a finite group 퐺 is said to be a 퐾- at the identity in the Cayley graph of the group and the approximate subgroup of 퐺 if 퐴 ⋅ 퐴 is contained in 퐾 (left) amazing fact represented in this theorem is that this geo- translates of 퐴. Approximate subgroups are generaliza- metric condition has serious algebraic consequences. (The tions of subgroups (since a 1-approximate group is simply converse of the theorem is also true and much easier to a subgroup of 퐺). The Freiman theorem for abelian groups prove.) classifies 퐾-approximate subgroups of abelian groups (they A key construction in the proof of Gromov’s theorem is are, in some sense, built from generalized arithmetic pro- that of an asymptotic cone of a metric space. Roughly speak- gressions and group extensions). It was an open problem ing, an asymptotic cone of a metric space is the result of whether or not a similar result held for 퐾-approximate sub- looking at the metric space from “very far away” retaining groups of arbitrary (not necessarily abelian) groups.

844 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 In [13], using sophisticated methods from model the- to the 휎-algebra generated by them. However, in nonstan- ory, Hrushovski made a major breakthrough in the above dard analysis, we build up more complicated sets from ba- so-called nonabelian Freiman problem. While he did not sic sets simultaneously in a structure and a nonstandard completely solve the problem, his breakthrough paved the extension, similar to how one might look at varieties both way for the work of Breuillard, Green, and Tao [4], who over a field and over various of its field extensions. Finally, reduced the use of sophisticated in favor of for our applications, we need a final (NA3) that en- mere nonstandard analysis (while using more intricate sures that our class of definable sets in the nonstandard combinatorics) and, in the process, succeeded in provid- extension has certain compactness properties. ing a complete classification of arbitrary approximate Structures and axiom (NA1). groups. The key insight of Hrushovski (which is also present Definition 6. A structure 퐒 = ((푆푖), (푅푗)) is a collection in the Breuillard–Green–Tao work) is that a nonstandard of sets (푆푖 ∶ 푖 ∈ 퐼), often called the basic sets of 퐒, to- (infinite) approximate group can be “modelled” by alo- gether with a collection (푅푗 ∶ 푗 ∈ 퐽) of distinguished cally compact group, which, by the structure theory of lo- relations on the basic sets, that is, for each 푗 ∈ 퐽, there cally compact groups, can in turn be modeled by finite- are 푖(1), … , 푖(푛) ∈ 퐼 such that 푅푗 ⊆ 푆푖(1) × ⋯ × 푆푖(푛). dimensional Lie groups. (This is not unrelated to the use Distinguished relations are also called primitives of 퐒. of nonstandard methods in the proof of Gromov’s theo- Later, it will be convenient to speak of the complete structure rem described above.) Thus, in attacking this problem of on (푆 ), which is simply the structure with basic sets (푆 ) finite combinatorics, one can use the infinitary tools from 푖 푖 and where we take all relations as the basic relations. differential geometry and Lie theory. Blurring the distinc- We can now state the first axiom of nonstandard exten- tion between the finite and the infinite is a cornerstone sions, which simply explains what makes them extensions: of many applications of nonstandard methods, allowing (NA1) Each basic set 푆 is extended to a set 푆∗ ⊇ 푆 and, one to leverage tools from either side of the divide when- 푖 푖 푖 to each distinguished relation 푅 as above, we associate a ever it proves convenient. Indeed, this is present even in 푗 corresponding relation the characterization of differentiability from Theorem 1, where the infinitary process of taking a limit is replaced by ∗ ∗ ∗ 푅푗 ⊆ 푆푖(1) × ⋯ × 푆푖(푛) the discretized finite quotient. We mention in passing that the structure theory of lo- whose intersection with 푆푖(1) × ⋯ × 푆푖(푛) is the original cally compact groups, due to Gleason, Montgomery, and relation 푅푗. Zippin, solved the fifth problem of Hilbert. Hirschfeld Definable sets and axiom (NA2). Let 퐒 be a structure. [12] used nonstandard analysis to give a conceptually sim- Some notation: given 푖 ∶= (푖(1), … , 푖(푛)), we set 푆푖 ∶= pler account of this solution. The first-named author gen- 푆푖(1) × ⋯ × 푆푖(푛). It will also be necessary to declare 푆∅ eralized Hirschfeld’s account to solve the local version of to be a one-element set. For the sake of readability, if 푖 Hilbert’s fifth problem [10]; this was, in turn, a part ofthe and 푗 are two finite sequences, then we write 푖 푗 for the Breuillard–Green–Tao work mentioned above. concatenation of the two sequences; if 푖 = (푖), then we 푖 푗 푗 Axioms for Nonstandard Extensions simply write and similarly for when is a one-element sequence. In this section, we describe an approach to nonstandard We define the collection of 퐒-definable sets to be the analysis using three axioms (NA1)-(NA3). This follows Boolean algebras 풟퐒(푖) (or simply 풟(푖) if 퐒 is clear from and elaborates on the treatment in the appendix of the context) of subsets of 푆 with the following properties: first-named author’s earlier coauthored work [7]. How- 푖 ever, when specialized to the real numbers, axioms (NA1)- (1) ∅, 푆푖 ∈ 풟(푖). (NA2) are very similar to Goldblatt’s [9] presentation of (2) For any 푖, {(푥, 푦) ∈ 푆푖 × 푆푖 ∶ 푥 = 푦} ∈ 풟(푖, 푖). nonstandard methods. (3) If 푎 ∈ 푆푖, then {푎} ∈ 풟(푖). The approach through (NA1)-(NA3) should remind the (4) If 푅푗 ⊆ 푆푖 is a basic relation, then 푅푗 ∈ 풟(푖). reader of the beginning of measure theory, where one sets (5) If 퐴 ∈ 풟(푖), then 퐴 × 푆푗 ∈ 풟(푖 푗) and 푆푗 × 퐴 ∈ out carefully a class of sets that form the building blocks of 풟(푗 푖). the subject-matter one is interested in. The most basic sets 퐴 ∈ 풟(푖) ⃗푖 = 푖 푗 푖 휋 ∶ 푆 → 푆 (6) If and 1 2 and 푖 푖1 푖2 is are those built into what we call the structures in (NA1)— the canonical projection, then 휋(퐴) ∈ 풟(푖1 푖2). and then we expand upon this class with the definable sets of axiom (NA2). This is similar to how, in measure the- While the closure properties of a Boolean algebra encode ory, one might pass from basic half-opens on the real line propositional operations such as conjunction and disjunc- tion, the additional postulates on the definable sets en- code the operations of predicate logic: the identity relation

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 845 is homophonically encoded in (2), while the last condi- the identity in 퐺. If 퐺 is first countable, then countable tion (6) encodes existential quantification since 푎1 푎2 is richness will ensure the existence of such elements. in the projection 휋(퐴) iff there is 푏 in 푆푗 with 푎1 푏 푎2 We should note that, for certain applications of non- belonging to 퐴. standard methods, countable richness is not a strong enough Now, if 퐷 ∈ 풟(푖) is an 퐒-definable set, then define assumption. For example, we may want the analogous ver- the 퐒∗-definable set 퐷∗ simply by replacing all instances sion of countable richness to hold for families of definable ∗ of basic relations 푅푗 used in the construction of 퐷 by 푅푗 . sets indexed by the real numbers rather than the natural Note that not every 퐒∗-definable set is of the form 퐷∗ for numbers. (Such richness is desirable, for example, in ap- ∗ an 퐒-definable set, e.g., {푎} for 푎 ∈ 푆푖 \푆푖. And we will plications to combinatorial number theory; see [6].) In soon see that such elements exist. this latter scenario, one would then assume that the non- We can now explain the second axiom of nonstandard standard extension is 픠-rich, where 픠 is the cardinality of analysis, the so-called , which in the ter- the real numbers. One can of course speak of greater as- minology of definable sets simply says that the property sumptions of richness, but we will refrain from dwelling of basic relations from (NA1) also holds for arbitrary 퐒- too much on this point. definable sets: This concludes the list of our axioms for nonstandard (NA2) For any 퐒-definable set 퐷 ∈ 풟(푖), we have 퐷∗∩ analysis. We remark that we could have also added distin- 푆푖 = 퐷. guished functions in our definition of a structure, and then Axiom (NA2) states precisely what it means for the non- a requirement in (NA2) for extensions of distinguished standard extension to be logically similar to the original functions. But working with graphs of functions, it is rou- structure. It is saying that any statement about elements tine to see that such a set-up can be encoded in a purely of the original structure that can be phrased in terms of de- relational structure as above. finable sets also holds within the nonstandard extension. We now stress: all applications of nonstandard analysis can Compactness and richness (NA3). Our final axiom asks proceed from the above three axioms alone. that our nonstandard extensions contain sufficiently many Power sets and internal sets. The reader already familiar “ideal” elements analogous to the infinitesimal elements with nonstandard analysis will notice that we have yet to added to 퐑∗ in “Calculus with Infinitesimals.” Here is the speak about a notion that is of central importance, namely precise statement: the notion of internal sets. We remedy that now. The structure 퐒 is said to be countably rich if for all ⃗푖 For certain basic sets 푆 of our structure 퐒, we often in- 풫(푆) and every countable family (푋푚)푚∈ℕ of elements of 풟(푖) clude also its as a basic set, and the mem- bership relation ∈푆∶= {(푥, 푌) ∈ 푆 × 풫(푆) ∶ 푥 ∈ 푌} with the finite intersection property, we have ⋂푚 푋푚 ≠ ∅. Here the “finite intersection property” means that as a primitive. This allows us to quantify over elements of 풫(푆), and thus gives enormous expressive power. For 푋푚1 ∩ ⋯ ∩ 푋푚푘 ≠ ∅ for all 푚1, … , 푚푘 ∈ ℕ. Countable richness can be seen as a logical compactness example, with 퐑 and 풫(퐑) as basic sets, together with the property when stated in its contraposed form: if 퐒 is count- membership relation between them and the ordering on 퐑 ably rich and an 퐒-definable set 푋 ∈ 풟(푖) is covered by as primitives, we can express by an elementary statement the fact that every nonempty subset of 퐑 with an upper countably many 퐒-definable sets 푋푚 ∈ 풟(푖), then 푋 is bound in 퐑 has a least upper bound in 퐑 (i.e., the com- already covered by finitely many of these 푋푚. Here is the final axiom of nonstandard extensions: pleteness of the real line). (NA3) 퐒∗ is countably rich. Given 푌 ∈ 풫(푆), the formula 푣 ∈푆 푌 (with 푣 a vari- Note that, as an ordered set, 퐑 is not countably rich, able ranging over 푆) defines the subset 푌 of 푆, so 푌 is not only an element in our structure 퐒, but also an 퐒- since ⋂푛(푛, +∞) = ∅. Thus our initial structure 퐒 will usually not be rich, which is why we consider now an ex- definable subset of 푆. In particular, every subset of 푆 is now 퐒∗ 퐒 퐒-definable (while not every subset of 풫(푆) is 퐒-definable). tension of such that (NA1), (NA2), and (NA3) hold, ∗ 퐒∗ 푋 Next, let an extension 퐒 of 퐒 be given that satisfies so is countably rich. One consequence is that if is ∗ ∗ 푋∗ (NA1) and (NA2). Then 푆 and 풫(푆) are basic sets of countable and infinite, then will have elements that ∗ ∗ are not in 푋. 퐒 , and the star extension ∈푆 of ∈푆 is among the prim- itives. There is no reason for the elements of 풫(푆)∗ to Richness is the precise formulation of asking that our ∗ ∗ nonstandard extension have certain ideal elements analo- be actual subsets of 푆 , or for ∈푆 to be an actual mem- 퐑∗ bership relation, but we always arrange this to be the case: gous to the infinitesimal elements of . For example, if ∗ ∗ ∗ 퐺 just replace 푃 ∈ 풫(푆) by the subset {푎 ∈ 푆 ∶ 푎 ∈푆 is a topological group, we will want there to be elements ∗ 푎 ∈ 퐺∗ 푃} of 푆 . This procedure is traditionally called Mostowski that are infinitely close to the identity element of ∗ ∗ 퐺 in the sense that 푎 ∈ 푈∗ for every neighborhood 푈 of collapse, and it identifies 풫(푆) with a subset of 풫(푆 ).

846 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 The subsets of 푆∗ that belong to 풫(푆)∗ via this identifica- We first note that there is 푚 ∈ ℕ such that 퐶∗ has only tion are traditionally called internal subsets of 푆∗. They are gaps of length at most 푚 on 퐼, whence 퐼 ⊆ 퐶∗ + [0, 푚]. ∗ ∗ in fact exactly the 퐒 -definable subsets of 푆 , as is easily Indeed, if this were not the case, then the set 푋푚 ∶= {푥 ∈ checked. 퐼 ∶ [푥, 푥+푚)∩퐶∗ = ∅} is a nonempty definable subset of ℕ∗ for each 푚, whence by countable saturation, there The Axioms in Action: Jin’s Theorem ∗ is 푥 ∈ ⋂푚 푋푚; we then have that 푥, 푥+1, 푥+2, … ∉ 퐶 , In this section, we give some details into the proof of Jin’s contradicting that 퐶∗ has only finite gaps on 퐼. theorem from “Jin’s Sumset Theorem.” Now, given 푘 ∈ ℕ, let 푌푘 ∶= {푥 ∈ ℕ ∶ [푥, 푥 + A few words on ℕ∗. First, it behooves us to give a pic- 푘) ⊆ 퐶 + [0, 푚]}. By the last paragraph and transfer, we ∗ ∗ ture of ℕ . We begin by noting that, by transfer, every have that 푌푘 ≠ ∅, whence, by transfer again, we have ∗ element 푁 of ℕ \ℕ is above every element ℕ in the or- that 푌푘 ≠ ∅. Since 푘 was arbitrary, this proves that 퐶 is dering. Transfer implies that any element 푁 in ℕ∗\ℕ has piecewise syndetic. □ a predecessor in ℕ∗, which we denote by 푁 − 1; clearly 푁 − 1 is also infinite, whence we may consider its prede- As an aside, we invite the reader to use the previous the- cessor 푁 − 2, another infinite element, and so on. Con- orem to give a quick nonstandard proof that the class of sequently, 푁 is contained in a copy of the integers (called piecewise syndetic subsets of ℕ is partition regular, mean- a ℤ-chain), the entirety of which is contained in the infi- ing that if 퐴 is piecewise syndetic and 퐴 = 퐵 ∪ 퐶, then ∗ nite part of ℕ . Note that 2푁 is an infinite number that at least one of 퐵 or 퐶 is piecewise syndetic. Partition reg- 푁 is contained in a strictly larger ℤ-chain, and that 2 is an ularity is a crucial notion in Ramsey theory, and the fact infinite number contained in a strictly smaller ℤ-chain. Fi- that the class of piecewise syndetic sets is partition regular 푁+푀 nally, note that if 푁 < 푀 are both infinite, then 2 is indicates that it is a robust structural notion of largeness. ∗ contained in a ℤ-chain strictly in-between the ℤ-chains de- For 푥, 푦 ∈ ℕ , we write 푥 ∼ℕ 푦 if and only if |푥 − 푦| termined by 푁 and 푀. We summarize: is finite. This is an equivalence relation on ℕ∗ and we set Lemma 7. The set of ℤ-chains determined by infinite elements 풞ℕ for the set of equivalence classes: per Lemma 7, this is of ℕ∗ is a dense linear order without endpoints. just the equivalence class of ℕ together with the ℤ-chains. ∗ ∗ We let 휋ℕ ∶ ℕ → 풞ℕ denote the canonical projection. By an interval 퐼 ⊆ ℕ , we mean a subset of the form ∗ Again by Lemma 7, the linear order on ℕ descends to a ∗ (푎, 푏) ∶= {푥 ∈ ℕ ∶ 푎 < 푥 < 푏} dense linear order on 풞ℕ. We can thus restate the previous for 푎 < 푏 in ℕ∗. The interval is said to be infinite if 푏 − 푎 theorem as: 퐶 is piecewise syndetic if and only if 휋ℕ(퐶) is infinite. contains an interval. Given a (finite) interval 퐼 ⊆ ℕ, every subset 퐷 of 퐼, Cuts. We now consider a different equivalence relation on ∗ ∗ being itself finite, of course has a cardinality that is anel- ℕ . Fix an infinite 푁 ∈ ℕ and consider instead the re- ∗ |푥−푦| ement of [0, |퐼|). Analogously, given an infinite interval lation ∼푁 on ℕ given by 푥 ∼푁 푦 if and only if 푁 ∗ 퐼 in ℕ and a definable subset 퐷 of 퐼, there is a natural is infinitesimal. We let 풞푁 denote the set of equivalence ∗ notion of the definable cardinality |퐷| of 퐷, which is an classes and 휋푁 ∶ ℕ → 풞푁 the canonical projection. This element of [0, |퐼|). Indeed, we can view the cardinality time, something interesting happens: as a linear order, an function | ⋅ | as a function on the product of basic sets initial segment of 풞푁 is isomorphic to the set of positive 풫(ℕ) × ℕ × ℕ given by | ⋅ |(퐷, 푎, 푏) ∶= |퐷 ∩ (푎, 푏)| reals (in particular, the initial segment consists of those 푥 if 푎 < 푏 (and some other default value otherwise); the 푥 such that 푁 is finite, and the isomorphism is given by 푥 nonstandard extension of this function then assigns a car- sending 푥 to st( 푁 )). ∗ dinality to definable subsets of intervals in ℕ . By transfer, The equivalence relations ∼ℕ and ∼푁 are instances of a ∗ the definable cardinality of an interval is simply its length. more general notion. We call an initial segment 푈 of ℕ a Piecewise syndeticity. Our next order of business is to cut if 푈 is closed under addition. Note that ℕ is the small- give a nonstandard description of piecewise syndeticity, est cut. Another example of a cut is the cut 푈푁 ∶= {퐾 ∈ ∗ 퐾 which we introduced in “Jin’s Sumset Theorem.” Given ℕ ∶ 푁 is infinitesimal}, where 푁 is infinite. As above, 퐶 ⊆ ℕ∗ 퐶 퐼 퐽 퐼 for a cut 푈, defining 푥 ∼푈 푦 if |푥 − 푦| ∈ 푈 yields an , a gap of on is a subinterval of such that ∗ 퐶 ∩ 퐽 = ∅. equivalence relation on ℕ with set of equivalence classes 풞푈 and projection map 휋푈. (We chose ∼푁 and 휋푁 as Theorem 8. 퐶 ⊆ ℕ is piecewise syndetic if and only if there is ∗ ∗ opposed to ∼푈푁 and 휋푈푁 for notational cleanliness.) As 퐼 ⊆ ℕ 퐶 ∗ an infinite interval such that has only finite gaps before, the usual order on ℕ descends to a linear order on 퐼. on 풞푈. Proof. We only prove the “if” direction. Suppose that 퐼 is We now recall a classical theorem of Steinhaus: if 퐸 and an infinite interval such that 퐶∗ has only finite gaps on 퐼. 퐹 are subsets of ℝ of positive Lebesgue measure, then 퐸+퐹

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 847 contains an interval. Given that reals are simply equiva- 퐼) − 푎 and 퐷 ∶= (퐵∗ ∩ 퐽) − 푏. Note then that 퐶 and 퐷 lence classes modulo the cut 푈푁 and in proving Jin’s theo- are definable subsets of [0, 푁) of positive Loeb measure. rem we are looking for sums of cuts modulo 푈ℕ to contain Without loss of generality, we may assume that 퐶 and 퐷 an interval, it raises the question as to whether or not there belong to the first half of [0, 푁). Then by Jin’s theorem is a natural measure on any cut space 풞푈 for which the mentioned above, 휋ℕ(퐶 + 퐷) contains an interval in 풞ℕ. analogue of Steinhaus’ theorem is true and which yields Translating back by 푎+푏, one finds an infinite hyperfinite ∗ ∗ Lebesgue measure in the case of the cut space 풞푁. interval on which 퐴 + 퐵 has only finite gaps, whence, Earlier, we saw that every definable subset of an interval by the nonstandard characterization described earlier, we ∗ 퐼 in ℕ has a definable cardinality. This procedure leads see that 퐴 + 퐵 is piecewise syndetic. to a natural measure 휇퐼 on the algebra of definable subsets |퐷| The Ultraproduct Construction of 퐼 given by 휇퐼(퐷) ∶= st( |퐼| ). The usual Carath´eodory extension procedure shows that 휇퐼 can be extended to a 휎- Of course, the lingering question remains: given a struc- additive probability measure on the 휎-algebra generated ture 퐒, is there a structure 퐒∗ satisfying (NA1)-(NA3)? by the definable subsets of 퐼. Indeed, countable richness of Model theorists know these axioms to be consistent by the nonstandard extension ensures that the hypotheses of basic model-theoretic facts. However, in this section, we the Carath´eodory’s extension theorem apply. The resulting present a construction that is much more “mainstream” measure is called the Loeb measure on 퐼 with corresponding and easy to describe to nonlogicians. 휎-algebra of Loeb measurable sets. Obtaining 퐑∗ as an ultrapower. As in the passage from If we are in the situation that 퐼 = [0, 푁) for 푁 infinite any number system to an extension where we are trying and 푈 is a cut contained in 퐼, then we can push forward to add desired elements (e.g., the passage from 퐍 to 퐙 to the Loeb measure to a probability measure on 풞푈, which 퐐 to 퐑 to 퐂), we simply formally add the new desired we also refer to as Loeb measure. elements and then see what technicalities we need to in- This procedure does in fact agree with Lebesgue mea- troduce to make this formal passage precise. In this case, ∗ sure in the case of the cuts 푈푁: in passing from 퐑 to 퐑 , we are trying to add infinite ele- 퐑 Theorem 9. Given an infinite 푁, the pushforward via 휋 of ments. Following the Cauchy sequence construction of 푁 퐐 1, 2, 3, … 퐑 the Loeb measure on [0, 푁) is the Lebesgue measure on [0, 1]. from , we can simply add the sequence to and view this sequence as an infinite element of 퐑∗. Of Motivated by the preceding discussion, Keisler and Leth course, just as in the case of the passage from 퐐 to 퐑, many asked whether the analog of Steinhaus’ result holds for ar- sequences should represent the same element of 퐑∗, e.g., bitrary cut spaces. Renling Jin answered this question affir- the sequence −32, 휋, 46, 4, 5, 6 … should represent the matively: same sequence as 1, 2, 3, …. In general, we should iden- Theorem 10 ([14]). Suppose that 퐼 = [0, 푁) is an infinite tify two sequences if they agree on a big number of indices, (푥 ) (푦 ) interval and 푈 is a cut contained in 퐼. If 퐴 and 퐵 are defin- where two sequences 푛 and 푛 agree on a big num- 푛 푥 = 푦 able subsets of 퐼 with positive Loeb measure, then 휋 (퐴 + 퐵) ber of terms if the set of for which 푛 푛 is a large 푈 퐍 contains an interval. subset of . Admittedly, the words “big” and “large” are rather vague here, so we need to isolate some properties Technically speaking, in order for this to literally be true, that large subsets of 퐍 should have; the resulting notion one needs to assume that 퐴 and 퐵 are contained in the first is that of a filter on 퐍. Since the definition makes perfect half of 퐼; otherwise, one needs to view 퐼 as a nonstandard sense for an arbitrary index set ℐ, we do so: cyclic group and then look at 퐴 + 퐵 in the group sense. Definition 11. Suppose that ℐ is a set. A (proper) filter on Without going into too much detail, the theorem is proven ℐ is a collection ℱ of subsets of ℐ satisfying: by contradiction, taking a “maximal” counterexample, and performing some nontrivial nonstandard counting. • ∅ ∉ ℱ, ℐ ∈ ℱ; • 퐴 ∈ ℱ 퐴 ⊆ 퐵 ⊆ ℐ 퐵 ∈ ℱ Finishing the proof of Jin’s theorem. We now have all of if and , then ; • 퐴, 퐵 ∈ ℱ 퐴 ∩ 퐵 ∈ ℱ the pieces necessary to prove the sumset theorem. Suppose if , then . that 퐴 and 퐵 are subsets of ℕ with positive Banach density. Denoting the set of ℐ-sequences by 퐑ℐ, the above defini- By the nonstandard characterization of limit, there is some tion of a filter was engineered so that the relation ∼ℱ on ℐ infinite interval 퐼 such that BD(퐴) is approximately equal 퐑 defined by 푥 ∼ℱ 푦 if and only if {푖 ∈ ℐ ∶ 푥푖 = 푦푖} ∈ |퐴∗∩퐼| ∗ to |퐼| . In other words, BD(퐴) = 휇퐼(퐴 ∩퐼). Likewise, ℱ is an equivalence relation. We denote the equivalence ∗ 퐼 there is an infinite interval 퐽 such that BD(퐵) = 휇퐽(퐵 ∩ class of an ℐ-sequence 푥 from 퐑 by [푥], and the set of all 퐽). Without loss of generality, we may assume that |퐼| = such equivalence classes by 퐑ℱ. By identifying an element |퐽| = 푁 for some infinite 푁. Let 푎 and 푏 denote the left 푟 ∈ 퐑 with the constant sequence 푐푟 = (푟, 푟, 푟, …), we endpoints of 퐼 and 퐽 respectively and set 퐶 ∶= (퐴∗ ∩ get a natural inclusion of 퐑 into 퐑ℐ, and by passing to the

848 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 equivalence class we obtain the natural inclusion of 퐑 into known as Łos’ theorem, which is easily proven by induction 퐑ℱ. In a diagram: on the “complexity” of definable sets. Further, that this construction works for any initial structure 퐒 is the fact 푟↦푐푟 / 푥↦[푥] / 퐑 퐑ℐ 퐑ℱ that one invokes to show that certain axiomatic theories of nonstandard methods are conservative over certain ax- 퐑ℱ Note also that has a natural field structure on it extend- iomatic theories of real numbers: i.e., anything that the 퐑 ing the field structure on . former theory proves the latter theory also proves (cf. [5]). There are now two problems with leaving things at this Getting richer. It turns out that obtaining countable rich- level of generality. The first can be motivated by the desire ness via ultrapowers is quite straightforward: to turn 퐑ℱ into an ordered field. The natural thing todo would be to declare [푥] < [푦] if and only if {푖 ∈ ℐ ∶ 푥푖 < Theorem 14 (Keisler [15]). Suppose that 풰 is a nonprincipal 푦푖} ∈ ℱ. However, it is entirely possible that, under the on a countably infinite index set ℐ. Then for any 풰 above definition, we may have ∼ℱ-inequivalent sequences structure 퐒, 퐒 is countably rich. 푥 and 푦 for which [푥] ≮ [푦] and [푦] ≮ [푥]. We can remedy this by adding one further requirement: What about higher richness? If one insists on only using as a means of producing nonstandard exten- Definition 12. A filter 풰 on ℐ is called an ultrafilter if, for sions, then one can indeed obtain nonstandard universes every 퐴 ⊆ ℐ, we have that either 퐴 ∈ 풰 or ℐ\퐴 ∈ 풰 (but with higher richness properties at the expense of dealing not both). with some messy infinite combinatorics. Indeed, Keisler isolated a combinatorial property of , called good- Here is the other problem: suppose that 풰37 is the col- ness, and proved the following theorem: lection of subsets of 퐍 defined by declaring 퐴 ∈ 풰37 if and only if 37 ∈ 퐴. Although this hardly matches the in- Theorem 15 (Keisler [16], [17]). Let 풰 be an ultrafilter on tuition of gathering large subsets of 퐍, it is easy to see that a set ℐ. Then 풰 is good if and only if: for every structure 퐒, 풰 풰37 is in fact an ultrafilter on 퐍, called the principal ultra- 퐒 is maximally rich. filter on 퐍 generated by 37. Now notice that in 퐑풰37 , every sequence is equivalent to the real number given by its 37th Here, we are using the admittedly vague term maximally entry. In other words, the infinite element 1, 2, 3, … that rich to mean that the structure is as rich as the cardinality we tried to add is not infinite at all, but rather is identified of its underlying domain allows. In order for this theo- with the very finite number 37. To avoid this triviality, we rem to be useful, one needs to know that good ultrafilters add the following requirement: exist. This is indeed the case: Keisler first proved, under the assumption of the Generalized Continuum Hypothe- Definition 13. If ℐ is an index set and 푖 ∈ 퐼, we call 풰푖 ∶= sis, that any infinite set possesses a good ultrafilter; later, {퐴 ⊆ 퐼 ∶ 푖 ∈ 퐴} the principal ultrafilter on 퐼 generated by Kunen proved this fact without any extra set-theoretic as- 푖. An ultrafilter 풰 on 퐼 is called principal if it is of the form sumption. 풰푖 for some 푖 ∈ 퐼 and is otherwise called nonprincipal. Other Approaches We may now summarize: for any nonprincipal ultrafil- Ever since its inception, a slew of different frameworks for 풰 ℐ 퐑풰 ter on any index set , is a proper ordered field exten- approaching nonstandard analysis have been presented. 퐑 퐼 sion of . Now, the ultrafilter on a set may also be viewed Many of these approaches can be viewed as attempts to {0, 1} ℐ as a finitely additive -valued measure on , which, axiomatize different aspects of the ultraproduct construc- 0 moreover, gives points measure precisely when the ultra- tion. In this section, we briefly describe the differences be- filter is nonprincipal. In this way, the ultrapower construc- tween these approaches and our preferred approach using 퐑 tion of bears much resemblance to the practice common (NA1)-(NA3). in measure theory of identifying measurable functions if One alternative approach is just to axiomatize the be- they agree almost everywhere. havior of the embedding of the original structure in its Ultrapowers of structures. The approach of the previous nonstandard extension. Depending on how formal one subsection works to obtain, given an initial structure 퐒, sought to be, one might then replace informal descriptions ∗ structures 퐒 satisfying (NA1) and (NA2). Indeed, given of the original structure by some axiomatic characteriza- ∗ 풰 any basic set 푆푖, we can set 푆푖 ∶= 푆푖 and given any ba- tion of it. However, as the ultraproduct construction it- ∗ 풰 ∗ 풰 sic relation 푅푗, we can set 푅푗 ∶= 푅푗 and 퐒 = 퐒 . In self shows, first-order axioms cannot describe infinite struc- this case, 퐒∗ is called the ultrapower rather than an ultra- tures up to isomorphism. Hence, if one sought first-order product. As before, we have the natural inclusions, and it axiomatic characterizations, one might seek to add on ax- is clear that 퐒∗ satisfies (NA1). The fact that (NA2) holds ioms suggestive of the way in which the original structure in this context is an instance of a result in model theory was more canonical than its nonstandard extension. For

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 849 instance, in the case of reals, one might add on first-order extensions of any structure, the approach works just the axioms to the effect that any bounded subset of 퐑 first- same for, e.g., nonstandard 푝-adic analysis as for nonstan- order definable by recourse to 퐑∗ had a least-upper bound dard real-analysis. By contrast, with 훼-theory, in each case (e.g., Nelson’s [19, 1166] principle of standardization). one has to isolate some basic axioms that suffice for the Similar expressive difficulties emerge when one tries to find derivation of the full transfer principle, and in each case first-order axiomatic renditions of the richness conditions one has to reprove the full transfer principle. (e.g., Nelson’s [19, 1166] principle of idealization, and Another recent approach due to Benci and di Nasso [2] Hrbacek’s principle of bounded idealization). Since (NA1) seeks to characterize the nonstandard extensions of the re- -(NA3) makes no pretense to be a first-order axiomatiza- als as images of certain rings (they also have similar re- tion, it can avoid these niceties and just be content with sults for certain classes of spaces). While they state their our informal understanding of the real numbers and the ring-theoretic results for the reals, their proof generalizes richness conditions. In this, (NA1)-(NA3) is similar to as follows (recall the notion of complete structure given the superstructure approach of Robinson–Zakon (cf. the immediately after Definition 6): Chang–Keisler model theory book), which additionally builds in the Mostowski collapse mentioned in “Power Sets Theorem 16. Suppose that 퐊 is the complete structure of an ∗ and Internal Sets” and adopts a formulation of richness uncountable ordered field. Then 퐊 satisfies (NA1)-(NA2) if ∗ which does not require the notion of definable set. and only if 퐊 is the image under a ring homomorphism of a A particularly influential axiomatic approach was that composable ring over 퐊. of Nelson’s theory ([19]). Nelson’s aim was In this, the ring is said to be composable over 퐊 if it is a sub- to create an overall set theory for nonstandard methods. ring of the ring 퐊ℐ of all functions from some index set ℐ In addition to the usual set-theoretic axioms, it also con- to the original field 퐊 which is closed under taking com- tained the aforementioned principles of standardization positions with functions from 퐊 to 퐊 (and which contains and idealization. Nelson’s approach seems natural given all the constant functions). However, this of course is not that much of mathematics can be replicated in set theory. to say that studying composable rings is always the best However, as we have sought to stress in the applications in way to study nonstandard methods. After all, any group “Selected Classical and Recent Applications,” many of the is the image of a free group, but it would be a mistake to most exciting applications of nonstandard analysis are to approach all problems in group theory through the lens of local areas of mathematics. Hence, as far as applications free groups. go, little seems to be gained by adopting the very global A unifying motivation behind these recent approaches perspective of set theory. is the desire for a more mathematically natural framework Another recent approach pioneered by Benci and di for nonstandard analysis. By contrast, Robinson himself Nasso [1] is to axiomatize the inclusion of the set of 퐑- ∗ was an instrumentalist about the methods he developed valued sequences into the ultraproduct 퐑 . Their theory is (cf. [5]): He thought that their worth was tied less to any called 훼-theory because they axiomatize the map 푥 ↦ [푥] notions of naturalness and more to their proven track record sending a sequence to its equivalence class (cf. the diagram in obtaining results about standard structures, such as we after Definition 11). They use the notation 푥 ↦ 푥[훼], have surveyed in “Selected Classical and Recent Applica- since this notation reminds one of field extensions. For in- tions.” As we have sought to emphasize throughout, non- stance, it follows from the ultraproduct construction that standard methods such as (NA1)-(NA3) are easy to state the operation sending 푥 to 푥[훼] commutes with addition and use, and their consistency is easily verifiable via the and multiplication: ultraproduct construction. In describing the ultraproduct 푥[훼]+푦[훼] = (푥+푦)[훼], 푥[훼]⋅푦[훼] = (푥⋅푦)[훼] construction we mentioned the analogy with the Lebesgue integral. Here is another respect in which they are simi- The idea of 훼-theory is to take these identities—and others lar: to use the Lebesgue integral correctly, one does not pertaining to sets of reals—as axioms, and to derive trans- need to keep its measure-theoretic construction constantly fer from these. The choice between 훼-theory and (NA1)- in view, but rather one can just work with characteristic (NA3) is similar then to the choice between Dedekind cuts properties of it, like the Dominated Convergence Theorem. and Cauchy sequences: they are both equally good descrip- Likewise, to use nonstandard methods correctly, one does tions of their subject matter, but they just differ in which not need to keep the ultraproduct construction constantly basic properties are derived and which are taken as primi- in mind. Rather, as we sought to illustrate with Jin’s Theo- tive. One reason traditionally given for preferring Cauchy rem in “The Axioms in Action: Jin’s Theorem” one can do sequences over Dedekind cuts is that it more readily gen- nonstandard analysis just using the three simple axioms eralizes to other situations, e.g., complete metric spaces. (NA1)-(NA3). Much the same is true of (NA1)-(NA3): since we can take

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