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Nonstandard analysis is a mathematical framework in which one extends the standard mathematical universe {{\mathfrak U}} of standard numbers, standard sets, standard functions, etc. into a larger nonstandard universe {{}^* {\mathfrak U}} of nonstandard numbers, nonstandard sets, nonstandard functions, etc., somewhat analogously to how one places the real numbers inside the complex numbers, or the rationals inside the reals. This nonstandard universe enjoys many of the same properties as the standard one; in particular, we have the transfer principle that asserts that any statement in the language of first order logic is true in the standard universe if and only if it is true in the nonstandard one. (For instance, because Fermat’s last theorem is known to be true for standard natural numbers, it is automatically true for nonstandard natural numbers as well.) However, the nonstandard universe also enjoys some additional useful properties that the standard one does not, most notably the countable saturation property, which is a property somewhat analogous to the completeness property of a metric space; much as metric completeness allows one to assert that the intersection of a countable family of nested closed balls is non-empty, countable saturation allows one to assert that the intersection of a countable family of nested satisfiable formulae is simultaneously satisfiable. (See this previous blog post for more on the analogy between the use of nonstandard analysis and the use of metric completions.) Furthermore, by viewing both the standard and nonstandard universes externally (placing them both inside a larger metatheory, such as a model of Zermelo-Frankel-Choice (ZFC) set theory; in some more advanced set-theoretic applications one may also wish to add some large cardinal axioms), one can place some useful additional definitions and constructions on these universes, such as defining the concept of an infinitesimal nonstandard number (a number which is smaller in magnitude than any positive standard number). The ability to rigorously manipulate infinitesimals is of course one of the most well-known advantages of working with nonstandard analysis.

To build a nonstandard universe {{}^* {\mathfrak U}} from a standard one {{\mathfrak U}}, the most common approach is to take an ultrapower of {{\mathfrak U}} with respect to some non-principal ultrafilter over the natural numbers; see e.g. this blog post for details. Once one is comfortable with ultrafilters and ultrapowers, this becomes quite a simple and elegant construction, and greatly demystifies the nature of nonstandard analysis.

On the other hand, nonprincipal ultrafilters do have some unappealing features. The most notable one is that their very existence requires the axiom of choice (or more precisely, a weaker form of this axiom known as the boolean prime ideal theorem). Closely related to this is the fact that one cannot actually write down any explicit example of a nonprincipal ultrafilter, but must instead rely on nonconstructive tools such as Zorn’s lemma, the Hahn-Banach theorem, Tychonoff’s theorem, the Stone-Cech compactification, or the boolean prime ideal theorem to locate one. As such, ultrafilters definitely belong to the “infinitary” side of mathematics, and one may feel that it is inappropriate to use such tools for “finitary” mathematical applications, such as those which arise in hard analysis. From a more practical viewpoint, because of the presence of the infinitary ultrafilter, it can be quite difficult (though usually not impossible, with sufficient patience and effort) to take a finitary result proven via nonstandard analysis and coax an effective quantitative bound from it.

There is however a “cheap” version of nonstandard analysis which is less powerful than the full version, but is not as infinitary in that it is constructive (in the sense of not requiring any sort of choice-type axiom), and which can be translated into standard analysis somewhat more easily than a fully nonstandard argument; indeed, a cheap nonstandard argument can often be presented (by judicious use of asymptotic notation) in a way which is nearly indistinguishable from a standard one. It is obtained by replacing the nonprincipal ultrafilter in fully nonstandard analysis with the more classical Fréchet filter of cofinite subsets of the natural numbers, which is the filter that implicitly underlies the concept of the classical limit {\lim_{{\bf n} \rightarrow \infty} a_{\bf n}} of a sequence when the underlying asymptotic parameter {{\bf n}} goes off to infinity. As such, “cheap nonstandard analysis” aligns very well with traditional mathematics, in which one often allows one’s objects to be parameterised by some external parameter such as {{\bf n}}, which is then allowed to approach some limit such as {\infty}. The catch is that the Fréchet filter is merely a filter and not an ultrafilter, and as such some of the key features of fully nonstandard analysis are lost. Most notably, the law of the excluded middle does not transfer over perfectly from standard analysis to cheap nonstandard analysis; much as there exist bounded sequences of real numbers (such as {0,1,0,1,\ldots}) which do not converge to a (classical) limit, there exist statements in cheap nonstandard analysis which are neither true nor false (at least without passing to a subsequence, see below). The loss of such a fundamental law of mathematical reasoning may seem like a major disadvantage for cheap nonstandard analysis, and it does indeed make cheap nonstandard analysis somewhat weaker than fully nonstandard analysis. But in some situations (particularly when one is reasoning in a “constructivist” or “intuitionistic” fashion, and in particular if one is avoiding too much reliance on set theory) it turns out that one can survive the loss of this law; and furthermore, the law of the excluded middle is still available for standard analysis, and so one can often proceed by working from time to time in the standard universe to temporarily take advantage of this law, and then transferring the results obtained there back to the cheap nonstandard universe once one no longer needs to invoke the law of the excluded middle. Furthermore, the law of the excluded middle can be recovered by adopting the freedom to pass to subsequences with regards to the asymptotic parameter {{\bf n}}; this technique is already in widespread use in the analysis of partial differential equations, although it is generally referred to by names such as “the compactness method” rather than as “cheap nonstandard analysis”.

Below the fold, I would like to describe this cheap version of nonstandard analysis, which I think can serve as a pedagogical stepping stone towards fully nonstandard analysis, as it is formally similar to (though weaker than) fully nonstandard analysis, but on the other hand is closer in practice to standard analysis. As we shall see below, the relation between cheap nonstandard analysis and standard analysis is analogous in many ways to the relation between probabilistic reasoning and deterministic reasoning; it also resembles somewhat the preference in much of modern mathematics for viewing mathematical objects as belonging to families (or to categories) to be manipulated en masse, rather than treating each object individually. (For instance, nonstandard analysis can be used as a partial substitute for scheme theory in order to obtain uniformly quantitative results in algebraic geometry, as discussed for instance in this previous blog post.)

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In the previous set of notes, we introduced the notion of an ultra approximate group – an ultraproduct {A = \prod_{n \rightarrow\alpha} A_n} of finite {K}-approximate groups {A_n} for some {K} independent of {n}, where each {K}-approximate group {A_n} may lie in a distinct ambient group {G_n}. Although these objects arise initially from the “finitary” objects {A_n}, it turns out that ultra approximate groups {A} can be profitably analysed by means of infinitary groups {L} (and in particular, locally compact groups or Lie groups {L}), by means of certain models {\rho: \langle A \rangle \rightarrow L} of {A} (or of the group {\langle A \rangle} generated by {A}). We will define precisely what we mean by a model later, but as a first approximation one can view a model as a representation of the ultra approximate group {A} (or of {\langle A \rangle}) that is “macroscopically faithful” in that it accurately describes the “large scale” behaviour of {A} (or equivalently, that the kernel of the representation is “microscopic” in some sense). In the next section we will see how one can use “Gleason lemma” technology to convert this macroscopic control of an ultra approximate group into microscopic control, which will be the key to classifying approximate groups.

Models of ultra approximate groups can be viewed as the multiplicative combinatorics analogue of the more well known concept of an ultralimit of metric spaces, which we briefly review below the fold as motivation.

The crucial observation is that ultra approximate groups enjoy a local compactness property which allows them to be usefully modeled by locally compact groups (and hence, through the Gleason-Yamabe theorem from previous notes, by Lie groups also). As per the Heine-Borel theorem, the local compactness will come from a combination of a completeness property and a local total boundedness property. The completeness property turns out to be a direct consequence of the countable saturation property of ultraproducts, thus illustrating one of the key advantages of the ultraproduct setting. The local total boundedness property is more interesting. Roughly speaking, it asserts that “large bounded sets” (such as {A} or {A^{100}}) can be covered by finitely many translates of “small bounded sets” {S}, where “small” is a topological group sense, implying in particular that large powers {S^m} of {S} lie inside a set such as {A} or {A^4}. The easiest way to obtain such a property comes from the following lemma of Sanders:

Lemma 1 (Sanders lemma) Let {A} be a finite {K}-approximate group in a (global) group {G}, and let {m \geq 1}. Then there exists a symmetric subset {S} of {A^4} with {|S| \gg_{K,m} |A|} containing the identity such that {S^m \subset A^4}.

This lemma has an elementary combinatorial proof, and is the key to endowing an ultra approximate group with locally compact structure. There is also a closely related lemma of Croot and Sisask which can achieve similar results, and which will also be discussed below. (The locally compact structure can also be established more abstractly using the much more general methods of definability theory, as was first done by Hrushovski, but we will not discuss this approach here.)

By combining the locally compact structure of ultra approximate groups {A} with the Gleason-Yamabe theorem, one ends up being able to model a large “ultra approximate subgroup” {A'} of {A} by a Lie group {L}. Such Lie models serve a number of important purposes in the structure theory of approximate groups. Firstly, as all Lie groups have a dimension which is a natural number, they allow one to assign a natural number “dimension” to ultra approximate groups, which opens up the ability to perform “induction on dimension” arguments. Secondly, Lie groups have an escape property (which is in fact equivalent to no small subgroups property): if a group element {g} lies outside of a very small ball {B_\epsilon}, then some power {g^n} of it will escape a somewhat larger ball {B_1}. Or equivalently: if a long orbit {g, g^2, \ldots, g^n} lies inside the larger ball {B_1}, one can deduce that the original element {g} lies inside the small ball {B_\epsilon}. Because all Lie groups have this property, we will be able to show that all ultra approximate groups {A} “essentially” have a similar property, in that they are “controlled” by a nearby ultra approximate group which obeys a number of escape-type properties analogous to those enjoyed by small balls in a Lie group, and which we will call a strong ultra approximate group. This will be discussed in the next set of notes, where we will also see how these escape-type properties can be exploited to create a metric structure on strong approximate groups analogous to the Gleason metrics studied in previous notes, which can in turn be exploited (together with an induction on dimension argument) to fully classify such approximate groups (in the finite case, at least).

There are some cases where the analysis is particularly simple. For instance, in the bounded torsion case, one can show that the associated Lie model {L} is necessarily zero-dimensional, which allows for a easy classification of approximate groups of bounded torsion.

Some of the material here is drawn from my recent paper with Ben Green and Emmanuel Breuillard, which is in turn inspired by a previous paper of Hrushovski.

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Roughly speaking, mathematical analysis can be divided into two major styles, namely hard analysis and soft analysis. The precise distinction between the two types of analysis is imprecise (and in some cases one may use a blend the two styles), but some key differences can be listed as follows.

  • Hard analysis tends to be concerned with quantitative or effective properties such as estimates, upper and lower bounds, convergence rates, and growth rates or decay rates. In contrast, soft analysis tends to be concerned with qualitative or ineffective properties such as existence and uniqueness, finiteness, measurability, continuity, differentiability, connectedness, or compactness.
  • Hard analysis tends to be focused on finitary, finite-dimensional or discrete objects, such as finite sets, finitely generated groups, finite Boolean combination of boxes or balls, or “finite-complexity” functions, such as polynomials or functions on a finite set. In contrast, soft analysis tends to be focused on infinitary, infinite-dimensional, or continuous objects, such as arbitrary measurable sets or measurable functions, or abstract locally compact groups.
  • Hard analysis tends to involve explicit use of many parameters such as {\epsilon}, {\delta}, {N}, etc. In contrast, soft analysis tends to rely instead on properties such as continuity, differentiability, compactness, etc., which implicitly are defined using a similar set of parameters, but whose parameters often do not make an explicit appearance in arguments.
  • In hard analysis, it is often the case that a key lemma in the literature is not quite optimised for the application at hand, and one has to reprove a slight variant of that lemma (using a variant of the proof of the original lemma) in order for it to be suitable for applications. In contrast, in soft analysis, key results can often be used as “black boxes”, without need of further modification or inspection of the proof.
  • The properties in soft analysis tend to enjoy precise closure properties; for instance, the composition or linear combination of continuous functions is again continuous, and similarly for measurability, differentiability, etc. In contrast, the closure properties in hard analysis tend to be fuzzier, in that the parameters in the conclusion are often different from the parameters in the hypotheses. For instance, the composition of two Lipschitz functions with Lipschitz constant {K} is still Lipschitz, but now with Lipschitz constant {K^2} instead of {K}. These changes in parameters mean that hard analysis arguments often require more “bookkeeping” than their soft analysis counterparts, and are less able to utilise algebraic constructions (e.g. quotient space constructions) that rely heavily on precise closure properties.

In the lectures so far, focusing on the theory surrounding Hilbert’s fifth problem, the results and techniques have fallen well inside the category of soft analysis. However, we will now turn to the theory of approximate groups, which is a topic which is traditionally studied using the methods of hard analysis. (Later we will also study groups of polynomial growth, which lies on an intermediate position in the spectrum between hard and soft analysis, and which can be profitably analysed using both styles of analysis.)

Despite the superficial differences between hard and soft analysis, though, there are a number of important correspondences between results in hard analysis and results in soft analysis. For instance, if one has some sort of uniform quantitative bound on some expression relating to finitary objects, one can often use limiting arguments to then conclude a qualitative bound on analogous expressions on infinitary objects, by viewing the latter objects as some sort of “limit” of the former objects. Conversely, if one has a qualitative bound on infinitary objects, one can often use compactness and contradiction arguments to recover uniform quantitative bounds on finitary objects as a corollary.

Remark 1 Another type of correspondence between hard analysis and soft analysis, which is “syntactical” rather than “semantical” in nature, arises by taking the proofs of a soft analysis result, and translating such a qualitative proof somehow (e.g. by carefully manipulating quantifiers) into a quantitative proof of an analogous hard analysis result. This type of technique is sometimes referred to as proof mining in the proof theory literature, and is discussed in this previous blog post (and its comments). We will however not employ systematic proof mining techniques here, although in later posts we will informally borrow arguments from infinitary settings (such as the methods used to construct Gleason metrics) and adapt them to finitary ones.

Let us illustrate the correspondence between hard and soft analysis results with a simple example.

Proposition 1 Let {X} be a sequentially compact topological space, let {S} be a dense subset of {X}, and let {f: X \rightarrow [0,+\infty]} be a continuous function (giving the extended half-line {[0,+\infty]} the usual order topology). Then the following statements are equivalent:

  • (i) (Qualitative bound on infinitary objects) For all {x \in X}, one has {f(x) < +\infty}.
  • (ii) (Quantitative bound on finitary objects) There exists {M < +\infty} such that {f(x) \leq M} for all {x \in S}.

In applications, {S} is typically a (non-compact) set of “finitary” (or “finite complexity”) objects of a certain class, and {X} is some sort of “completion” or “compactification” of {S} which admits additional “infinitary” objects that may be viewed as limits of finitary objects.

Proof: To see that (ii) implies (i), observe from density that every point {x} in {X} is adherent to {S}, and so given any neighbourhood {U} of {x}, there exists {y \in S \cap U}. Since {f(y) \leq M}, we conclude from the continuity of {f} that {f(x) \leq M} also, and the claim follows.

Conversely, to show that (i) implies (ii), we use the “compactness and contradiction” argument. Suppose for sake of contradiction that (ii) failed. Then for any natural number {n}, there exists {x_n \in S} such that {f(x_n) \geq n}. (Here we have used the axiom of choice, which we will assume throughout this course.) Using sequential compactness, and passing to a subsequence if necessary, we may assume that the {x_n} converge to a limit {x \in X}. By continuity of {f}, this implies that {f(x) = +\infty}, contradicting (i). \Box

Remark 2 Note that the above deduction of (ii) from (i) is ineffective in that it gives no explicit bound on the uniform bound {M} in (ii). Without any further information on how the qualitative bound (i) is proven, this is the best one can do in general (and this is one of the most significant weaknesses of infinitary methods when used to solve finitary problems); but if one has access to the proof of (i), one can often finitise or proof mine that argument to extract an effective bound for {M}, although often the bound one obtains in the process is quite poor (particularly if the proof of (i) relied extensively on infinitary tools, such as limits). See this blog post for some related discussion.

The above simple example illustrates that in order to get from an “infinitary” statement such as (i) to a “finitary” statement such as (ii), a key step is to be able to take a sequence {(x_n)_{n \in {\bf N}}} (or in some cases, a more general net {(x_\alpha)_{\alpha \in A}}) of finitary objects and extract a suitable infinitary limit object {x}. In the literature, there are three main ways in which one can extract such a limit:

  • (Topological limit) If the {x_n} are all elements of some topological space {S} (e.g. an incomplete function space) which has a suitable “compactification” or “completion” {X} (e.g. a Banach space), then (after passing to a subsequence if necessary) one can often ensure the {x_n} converge in a topological sense (or in a metrical sense) to a limit {x}. The use of this type of limit to pass between quantitative/finitary and qualitative/infinitary results is particularly common in the more analytical areas of mathematics (such as ergodic theory, asymptotic combinatorics, or PDE), due to the abundance of useful compactness results in analysis such as the (sequential) Banach-Alaoglu theorem, Prokhorov’s theorem, the Helly selection theorem, the Arzelá-Ascoli theorem, or even the humble Bolzano-Weierstrass theorem. However, one often has to take care with the nature of convergence, as many compactness theorems only guarantee convergence in a weak sense rather than in a strong one.
  • (Categorical limit) If the {x_n} are all objects in some category (e.g. metric spaces, groups, fields, etc.) with a number of morphisms between the {x_n} (e.g. morphisms from {x_{n+1}} to {x_n}, or vice versa), then one can often form a direct limit {\lim_{\rightarrow} x_n} or inverse limit {\lim_{\leftarrow} x_n} of these objects to form a limiting object {x}. The use of these types of limits to connect quantitative and qualitative results is common in subjects such as algebraic geometry that are particularly amenable to categorical ways of thinking. (We have seen inverse limits appear in the discussion of Hilbert’s fifth problem, although in that context they were not really used to connect quantitative and qualitative results together.)
  • (Logical limit) If the {x_n} are all distinct spaces (or elements or subsets of distinct spaces), with few morphisms connecting them together, then topological and categorical limits are often unavailable or unhelpful. In such cases, however, one can still tie together such objects using an ultraproduct construction (or similar device) to create a limiting object {\lim_{n \rightarrow \alpha} x_n} or limiting space {\prod_{n \rightarrow \alpha} x_n} that is a logical limit of the {x_n}, in the sense that various properties of the {x_n} (particularly those that can be phrased using the language of first-order logic) are preserved in the limit. As such, logical limits are often very well suited for the task of connecting finitary and infinitary mathematics together. Ultralimit type constructions are of course used extensively in logic (particularly in model theory), but are also popular in metric geometry. They can also be used in many of the previously mentioned areas of mathematics, such as algebraic geometry (as discussed in this previous post).

The three types of limits are analogous in many ways, with a number of connections between them. For instance, in the study of groups of polynomial growth, both topological limits (using the metric notion of Gromov-Hausdorff convergence) and logical limits (using the ultralimit construction) are commonly used, and to some extent the two constructions are at least partially interchangeable in this setting. (See also these previous posts for the use of ultralimits as a substitute for topological limits.) In the theory of approximate groups, though, it was observed by Hrushovski that logical limits (and in particular, ultraproducts) are the most useful type of limit to connect finitary approximate groups to their infinitary counterparts. One reason for this is that one is often interested in obtaining results on approximate groups {A} that are uniform in the choice of ambient group {G}. As such, one often seeks to take a limit of approximate groups {A_n} that lie in completely unrelated ambient groups {G_n}, with no obvious morphisms or metrics tying the {G_n} to each other. As such, the topological and categorical limits are not easily usable, whereas the logical limits can still be employed without much difficulty.

Logical limits are closely tied with non-standard analysis. Indeed, by applying an ultraproduct construction to standard number systems such as the natural numbers {{\bf N}} or the reals {{\bf R}}, one can obtain nonstandard number systems such as the nonstandard natural numbers {{}^* {\bf N}} or the nonstandard real numbers (or hyperreals) {{}^* {\bf R}}. These nonstandard number systems behave very similarly to their standard counterparts, but also enjoy the advantage of containing the standard number systems as proper subsystems (e.g. {{\bf R}} is a subring of {{}^* {\bf R}}), which allows for some convenient algebraic manipulations (such as the quotient space construction to create spaces such as {{}^* {\bf R} / {\bf R}}) which are not easily accessible in the purely standard universe. Nonstandard spaces also enjoy a useful completeness property, known as countable saturation, which is analogous to metric completeness (as discussed in this previous blog post) and which will be particularly useful for us in tying together the theory of approximate groups with the theory of Hilbert’s fifth problem. See this previous post for more discussion on ultrafilters and nonstandard analysis.

In these notes, we lay out the basic theory of ultraproducts and ultralimits (in particular, proving Los’s theorem, which roughly speaking asserts that ultralimits are limits in a logical sense, as well as the countable saturation property alluded to earlier). We also lay out some of the basic foundations of nonstandard analysis, although we will not rely too heavily on nonstandard tools in this course. Finally, we apply this general theory to approximate groups, to connect finite approximate groups to an infinitary type of approximate group which we will call an ultra approximate group. We will then study these ultra approximate groups (and models of such groups) in more detail in the next set of notes.

Remark 3 Throughout these notes (and in the rest of the course), we will assume the axiom of choice, in order to easily use ultrafilter-based tools. If one really wanted to expend the effort, though, one could eliminate the axiom of choice from the proofs of the final “finitary” results that one is ultimately interested in proving, at the cost of making the proofs significantly lengthier. Indeed, there is a general result of Gödel that any result which can be stated in the language of Peano arithmetic (which, roughly speaking, means that the result is “finitary” in nature), and can be proven in set theory using the axiom of choice (or more precisely, in the ZFC axiom system), can also be proven in set theory without the axiom of choice (i.e. in the ZF system). As this course is not focused on foundations, we shall simply assume the axiom of choice henceforth to avoid further distraction by such issues.

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This fall (starting Monday, September 26), I will be teaching a graduate topics course which I have entitled “Hilbert’s fifth problem and related topics.” The course is going to focus on three related topics:

  • Hilbert’s fifth problem on the topological description of Lie groups, as well as the closely related (local) classification of locally compact groups (the Gleason-Yamabe theorem).
  • Approximate groups in nonabelian groups, and their classification via the Gleason-Yamabe theorem (this is very recent work of Emmanuel Breuillard, Ben Green, Tom Sanders, and myself, building upon earlier work of Hrushovski);
  • Gromov’s theorem on groups of polynomial growth, as proven via the classification of approximate groups (as well as some consequences to fundamental groups of Riemannian manifolds).

I have already blogged about these topics repeatedly in the past (particularly with regard to Hilbert’s fifth problem), and I intend to recycle some of that material in the lecture notes for this course.

The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):

  • (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
  • (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)

Let me illustrate what I mean by these two principles with two simple examples, one in the continuous setting and one in the discrete setting. We begin with a continuous example. Given an {n \times n} complex matrix {A \in M_n({\bf C})}, define the matrix exponential {\exp(A)} of {A} by the formula

\displaystyle  \exp(A) := \sum_{k=0}^\infty \frac{A^k}{k!} = 1 + A + \frac{1}{2!} A^2 + \frac{1}{3!} A^3 + \ldots

which can easily be verified to be an absolutely convergent series.

Exercise 1 Show that the map {A \mapsto \exp(A)} is a real analytic (and even complex analytic) map from {M_n({\bf C})} to {M_n({\bf C})}, and obeys the restricted homomorphism property

\displaystyle  \exp(sA) \exp(tA) = \exp((s+t)A) \ \ \ \ \ (1)

for all {A \in M_n({\bf C})} and {s,t \in {\bf C}}.

Proposition 1 (Rigidity and structure of matrix homomorphisms) Let {n} be a natural number. Let {GL_n({\bf C})} be the group of invertible {n \times n} complex matrices. Let {\Phi: {\bf R} \rightarrow GL_n({\bf C})} be a map obeying two properties:

  • (Group-like object) {\Phi} is a homomorphism, thus {\Phi(s) \Phi(t) = \Phi(s+t)} for all {s,t \in {\bf R}}.
  • (Weak regularity) The map {t \mapsto \Phi(t)} is continuous.


  • (Strong regularity) The map {t \mapsto \Phi(t)} is smooth (i.e. infinitely differentiable). In fact it is even real analytic.
  • (Lie-type structure) There exists a (unique) complex {n \times n} matrix {A} such that {\Phi(t) = \exp(tA)} for all {t \in {\bf R}}.

Proof: Let {\Phi} be as above. Let {\epsilon > 0} be a small number (depending only on {n}). By the homomorphism property, {\Phi(0) = 1} (where we use {1} here to denote the identity element of {GL_n({\bf C})}), and so by continuity we may find a small {t_0>0} such that {\Phi(t) = 1 + O(\epsilon)} for all {t \in [-t_0,t_0]} (we use some arbitrary norm here on the space of {n \times n} matrices, and allow implied constants in the {O()} notation to depend on {n}).

The map {A \mapsto \exp(A)} is real analytic and (by the inverse function theorem) is a diffeomorphism near {0}. Thus, by the inverse function theorem, we can (if {\epsilon} is small enough) find a matrix {B} of size {B = O(\epsilon)} such that {\Phi(t_0) = \exp(B)}. By the homomorphism property and (1), we thus have

\displaystyle  \Phi(t_0/2)^2 = \Phi(t_0) = \exp(B) = \exp(B/2)^2.

On the other hand, by another application of the inverse function theorem we see that the squaring map {A \mapsto A^2} is a diffeomorphism near {1} in {GL_n({\bf C})}, and thus (if {\epsilon} is small enough)

\displaystyle  \Phi(t_0/2) = \exp(B/2).

We may iterate this argument (for a fixed, but small, value of {\epsilon}) and conclude that

\displaystyle  \Phi(t_0/2^k) = \exp(B/2^k)

for all {k = 0,1,2,\ldots}. By the homomorphism property and (1) we thus have

\displaystyle  \Phi(qt_0) = \exp(qB)

whenever {q} is a dyadic rational, i.e. a rational of the form {a/2^k} for some integer {a} and natural number {k}. By continuity we thus have

\displaystyle  \Phi(st_0) = \exp(sB)

for all real {s}. Setting {A := B/t_0} we conclude that

\displaystyle  \Phi(t) = \exp(tA)

for all real {t}, which gives existence of the representation and also real analyticity and smoothness. Finally, uniqueness of the representation {\Phi(t) = \exp(tA)} follows from the identity

\displaystyle  A = \frac{d}{dt} \exp(tA)|_{t=0}.


Exercise 2 Generalise Proposition 1 by replacing the hypothesis that {\Phi} is continuous with the hypothesis that {\Phi} is Lebesgue measurable (Hint: use the Steinhaus theorem.). Show that the proposition fails (assuming the axiom of choice) if this hypothesis is omitted entirely.

Note how one needs both the group-like structure and the weak regularity in combination in order to ensure the strong regularity; neither is sufficient on its own. We will see variants of the above basic argument throughout the course. Here, the task of obtaining smooth (or real analytic structure) was relatively easy, because we could borrow the smooth (or real analytic) structure of the domain {{\bf R}} and range {M_n({\bf C})}; but, somewhat remarkably, we shall see that one can still build such smooth or analytic structures even when none of the original objects have any such structure to begin with.

Now we turn to a second illustration of the above principles, namely Jordan’s theorem, which uses a discreteness hypothesis to upgrade Lie type structure to nilpotent (and in this case, abelian) structure. We shall formulate Jordan’s theorem in a slightly stilted fashion in order to emphasise the adherence to the above-mentioned principles.

Theorem 2 (Jordan’s theorem) Let {G} be an object with the following properties:

  • (Group-like object) {G} is a group.
  • (Discreteness) {G} is finite.
  • (Lie-type structure) {G} is contained in {U_n({\bf C})} (the group of unitary {n \times n} matrices) for some {n}.

Then there is a subgroup {G'} of {G} such that

  • ({G'} is close to {G}) The index {|G/G'|} of {G'} in {G} is {O_n(1)} (i.e. bounded by {C_n} for some quantity {C_n} depending only on {n}).
  • (Nilpotent-type structure) {G'} is abelian.

A key observation in the proof of Jordan’s theorem is that if two unitary elements {g, h \in U_n({\bf C})} are close to the identity, then their commutator {[g,h] = g^{-1}h^{-1}gh} is even closer to the identity (in, say, the operator norm {\| \|_{op}}). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have

\displaystyle  \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}

\displaystyle  = \| (g-1)(h-1) - (h-1)(g-1) \|_{op}

and so by the triangle inequality

\displaystyle  \| [g,h] - 1 \|_{op} \leq 2 \|g-1\|_{op} \|h-1\|_{op}. \ \ \ \ \ (2)

Now we can prove Jordan’s theorem.

Proof: We induct on {n}, the case {n=1} being trivial. Suppose first that {G} contains a central element {g} which is not a multiple of the identity. Then, by definition, {G} is contained in the centraliser {Z(g)} of {g}, which by the spectral theorem is isomorphic to a product {U_{n_1}({\bf C}) \times \ldots \times U_{n_k}({\bf C})} of smaller unitary groups. Projecting {G} to each of these factor groups and applying the induction hypothesis, we obtain the claim.

Thus we may assume that {G} contains no central elements other than multiples of the identity. Now pick a small {\epsilon > 0} (one could take {\epsilon=\frac{1}{10d}} in fact) and consider the subgroup {G'} of {G} generated by those elements of {G} that are within {\epsilon} of the identity (in the operator norm). By considering a maximal {\epsilon}-net of {G} we see that {G'} has index at most {O_{n,\epsilon}(1)} in {G}. By arguing as before, we may assume that {G'} has no central elements other than multiples of the identity.

If {G'} consists only of multiples of the identity, then we are done. If not, take an element {g} of {G'} that is not a multiple of the identity, and which is as close as possible to the identity (here is where we crucially use that {G} is finite). By (2), we see that if {\epsilon} is sufficiently small depending on {n}, and if {h} is one of the generators of {G'}, then {[g,h]} lies in {G'} and is closer to the identity than {g}, and is thus a multiple of the identity. On the other hand, {[g,h]} has determinant {1}. Given that it is so close to the identity, it must therefore be the identity (if {\epsilon} is small enough). In other words, {g} is central in {G'}, and is thus a multiple of the identity. But this contradicts the hypothesis that there are no central elements other than multiples of the identity, and we are done. \Box

Commutator estimates such as (2) will play a fundamental role in many of the arguments we will see in this course; as we saw above, such estimates combine very well with a discreteness hypothesis, but will also be very useful in the continuous setting.

Exercise 3 Generalise Jordan’s theorem to the case when {G} is a finite subgroup of {GL_n({\bf C})} rather than of {U_n({\bf C})}. (Hint: The elements of {G} are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of {{\bf C}^n}. However, if one averages that inner product by the finite group {G}, one obtains a new inner product on {{\bf C}^n} that is preserved by {G}, which allows one to conjugate {G} to a subgroup of {U_n({\bf C})}. This averaging trick is (a small) part of Weyl’s unitary trick in representation theory.)

Exercise 4 (Inability to discretise nonabelian Lie groups) Show that if {n \geq 3}, then the orthogonal group {O_n({\bf R})} cannot contain arbitrarily dense finite subgroups, in the sense that there exists an {\epsilon = \epsilon_n > 0} depending only on {n} such that for every finite subgroup {G} of {O_n({\bf R})}, there exists a ball of radius {\epsilon} in {O_n({\bf R})} (with, say, the operator norm metric) that is disjoint from {G}. What happens in the {n=2} case?

Remark 1 More precise classifications of the finite subgroups of {U_n({\bf C})} are known, particularly in low dimensions. For instance, one can show that the only finite subgroups of {SO_3({\bf R})} (which {SU_2({\bf C})} is a double cover of) are isomorphic to either a cyclic group, a dihedral group, or the symmetry group of one of the Platonic solids.

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I have blogged several times in the past about nonstandard analysis, which among other things is useful in allowing one to import tools from infinitary (or qualitative) mathematics in order to establish results in finitary (or quantitative) mathematics. One drawback, though, to using nonstandard analysis methods is that the bounds one obtains by such methods are usually ineffective: in particular, the conclusions of a nonstandard analysis argument may involve an unspecified constant {C} that is known to be finite but for which no explicit bound is obviously available. (In many cases, a bound can eventually be worked out by performing proof mining on the argument, and in particular by carefully unpacking the proofs of all the various results from infinitary mathematics that were used in the argument, as opposed to simply using them as “black boxes”, but this is a time-consuming task and the bounds that one eventually obtains tend to be quite poor (e.g. tower exponential or Ackermann type bounds are not uncommon).)

Because of this fact, it would seem that quantitative bounds, such as polynomial type bounds {X \leq C Y^C} that show that one quantity {X} is controlled in a polynomial fashion by another quantity {Y}, are not easily obtainable through the ineffective methods of nonstandard analysis. Actually, this is not the case; as I will demonstrate by an example below, nonstandard analysis can certainly yield polynomial type bounds. The catch is that the exponent {C} in such bounds will be ineffective; but nevertheless such bounds are still good enough for many applications.

Let us now illustrate this by reproving a lemma from this paper of Mei-Chu Chang (Lemma 2.14, to be precise), which was recently pointed out to me by Van Vu. Chang’s paper is focused primarily on the sum-product problem, but she uses a quantitative lemma from algebraic geometry which is of independent interest. To motivate the lemma, let us first establish a qualitative version:

Lemma 1 (Qualitative solvability) Let {P_1,\ldots,P_r: {\bf C}^d \rightarrow {\bf C}} be a finite number of polynomials in several variables with rational coefficients. If there is a complex solution {z = (z_1,\ldots,z_d) \in {\bf C}^d} to the simultaneous system of equations

\displaystyle  P_1(z) = \ldots = P_r(z) = 0,

then there also exists a solution {z \in \overline{{\bf Q}}^d} whose coefficients are algebraic numbers (i.e. they lie in the algebraic closure {{\bf Q}} of the rationals).

Proof: Suppose there was no solution to {P_1(z)=\ldots=P_r(z)=0} over {\overline{{\bf Q}}}. Applying Hilbert’s nullstellensatz (which is available as {\overline{{\bf Q}}} is algebraically closed), we conclude the existence of some polynomials {Q_1,\ldots,Q_r} (with coefficients in {\overline{{\bf Q}}}) such that

\displaystyle  P_1 Q_1 + \ldots + P_r Q_r = 1

as polynomials. In particular, we have

\displaystyle  P_1(z) Q_1(z) + \ldots + P_r(z) Q_r(z) = 1

for all {z \in {\bf C}^d}. This shows that there is no solution to {P_1(z)=\ldots=P_r(z)=0} over {{\bf C}}, as required. \Box

Remark 1 Observe that in the above argument, one could replace {{\bf Q}} and {{\bf C}} by any other pair of fields, with the latter containing the algebraic closure of the former, and still obtain the same result.

The above lemma asserts that if a system of rational equations is solvable at all, then it is solvable with some algebraic solution. But it gives no bound on the complexity of that solution in terms of the complexity of the original equation. Chang’s lemma provides such a bound. If {H \geq 1} is an integer, let us say that an algebraic number has height at most {H} if its minimal polynomial (after clearing denominators) consists of integers of magnitude at most {H}.

Lemma 2 (Quantitative solvability) Let {P_1,\ldots,P_r: {\bf C}^d \rightarrow {\bf C}} be a finite number of polynomials of degree at most {D} with rational coefficients, each of height at most {H}. If there is a complex solution {z = (z_1,\ldots,z_d) \in {\bf C}^d} to the simultaneous system of equations

\displaystyle  P_1(z) = \ldots = P_r(z) = 0,

then there also exists a solution {z \in \overline{{\bf Q}}^d} whose coefficients are algebraic numbers of degree at most {C} and height at most {CH^C}, where {C = C_{D, d,r}} depends only on {D}, {d} and {r}.

Chang proves this lemma by essentially establishing a quantitative version of the nullstellensatz, via elementary elimination theory (somewhat similar, actually, to the approach I took to the nullstellensatz in my own blog post). She also notes that one could also establish the result through the machinery of Gröbner bases. In each of these arguments, it was not possible to use Lemma 1 (or the closely related nullstellensatz) as a black box; one actually had to unpack one of the proofs of that lemma or nullstellensatz to get the polynomial bound. However, using nonstandard analysis, it is possible to get such polynomial bounds (albeit with an ineffective value of the constant {C}) directly from Lemma 1 (or more precisely, the generalisation in Remark 1) without having to inspect the proof, and instead simply using it as a black box, thus providing a “soft” proof of Lemma 2 that is an alternative to the “hard” proofs mentioned above.

Here’s how the proof works. Informally, the idea is that Lemma 2 should follow from Lemma 1 after replacing the field of rationals {{\bf Q}} with “the field of rationals of polynomially bounded height”. Unfortunately, the latter object does not really make sense as a field in standard analysis; nevertheless, it is a perfectly sensible object in nonstandard analysis, and this allows the above informal argument to be made rigorous.

We turn to the details. As is common whenever one uses nonstandard analysis to prove finitary results, we use a “compactness and contradiction” argument (or more precisely, an “ultralimit and contradiction” argument). Suppose for contradiction that Lemma 2 failed. Carefully negating the quantifiers (and using the axiom of choice), we conclude that there exists {D, d, r} such that for each natural number {n}, there is a positive integer {H^{(n)}} and a family {P_1^{(n)}, \ldots, P_r^{(n)}: {\bf C}^d \rightarrow {\bf C}} of polynomials of degree at most {D} and rational coefficients of height at most {H^{(n)}}, such that there exist at least one complex solution {z^{(n)} \in {\bf C}^d} to

\displaystyle  P_1^{(n)}(z^{(n)}) = \ldots = P_r(z^{(n)}) = 0, \ \ \ \ \ (1)

but such that there does not exist any such solution whose coefficients are algebraic numbers of degree at most {n} and height at most {n (H^{(n)})^n}.

Now we take ultralimits (see e.g. this previous blog post of a quick review of ultralimit analysis, which we will assume knowledge of in the argument that follows). Let {p \in \beta {\bf N} \backslash {\bf N}} be a non-principal ultrafilter. For each {i=1,\ldots,r}, the ultralimit

\displaystyle  P_i := \lim_{n \rightarrow p} P_i^{(n)}

of the (standard) polynomials {P_i^{(n)}} is a nonstandard polynomial {P_i: {}^* {\bf C}^d \rightarrow {}^* {\bf C}} of degree at most {D}, whose coefficients now lie in the nonstandard rationals {{}^* {\bf Q}}. Actually, due to the height restriction, we can say more. Let {H := \lim_{n \rightarrow p} H^{(n)} \in {}^* {\bf N}} be the ultralimit of the {H^{(n)}}, this is a nonstandard natural number (which will almost certainly be unbounded, but we will not need to use this). Let us say that a nonstandard integer {a} is of polynomial size if we have {|a| \leq C H^C} for some standard natural number {C}, and say that a nonstandard rational number {a/b} is of polynomial height if {a}, {b} are of polynomial size. Let {{\bf Q}_{poly(H)}} be the collection of all nonstandard rationals of polynomial height. (In the language of nonstandard analysis, {{\bf Q}_{poly(H)}} is an external set rather than an internal one, because it is not itself an ultraproduct of standard sets; but this will not be relevant for the argument that follows.) It is easy to see that {{\bf Q}_{poly(H)}} is a field, basically because the sum or product of two integers of polynomial size, remains of polynomial size. By construction, it is clear that the coefficients of {P_i} are nonstandard rationals of polynomial height, and thus {P_1,\ldots,P_r} are defined over {{\bf Q}_{poly(H)}}.

Meanwhile, if we let {z := \lim_{n \rightarrow p} z^{(n)} \in {}^* {\bf C}^d} be the ultralimit of the solutions {z^{(n)}} in (1), we have

\displaystyle  P_1(z) = \ldots = P_r(z) = 0,

thus {P_1,\ldots,P_r} are solvable in {{}^* {\bf C}}. Applying Lemma 1 (or more precisely, the generalisation in Remark 1), we see that {P_1,\ldots,P_r} are also solvable in {\overline{{\bf Q}_{poly(H)}}}. (Note that as {{\bf C}} is algebraically closed, {{}^*{\bf C}} is also (by Los’s theorem), and so {{}^* {\bf C}} contains {\overline{{\bf Q}_{poly(H)}}}.) Thus, there exists {w \in \overline{{\bf Q}_{poly(H)}}^d} with

\displaystyle  P_1(w) = \ldots = P_r(w) = 0.

As {\overline{{\bf Q}_{poly(H)}}^d} lies in {{}^* {\bf C}^d}, we can write {w} as an ultralimit {w = \lim_{n \rightarrow p} w^{(n)}} of standard complex vectors {w^{(n)} \in {\bf C}^d}. By construction, the coefficients of {w} each obey a non-trivial polynomial equation of degree at most {C} and whose coefficients are nonstandard integers of magnitude at most {C H^C}, for some standard natural number {C}. Undoing the ultralimit, we conclude that for {n} sufficiently close to {p}, the coefficients of {w^{(n)}} obey a non-trivial polynomial equation of degree at most {C} whose coefficients are standard integers of magnitude at most {C (H^{(n)})^C}. In particular, these coefficients have height at most {C (H^{(n)})^C}. Also, we have

\displaystyle  P_1^{(n)}(w^{(n)}) = \ldots = P_r^{(n)}(w^{(n)}) = 0.

But for {n} larger than {C}, this contradicts the construction of the {P_i^{(n)}}, and the claim follows. (Note that as {p} is non-principal, any neighbourhood of {p} in {{\bf N}} will contain arbitrarily large natural numbers.)

Remark 2 The same argument actually gives a slightly stronger version of Lemma 2, namely that the integer coefficients used to define the algebraic solution {z} can be taken to be polynomials in the coefficients of {P_1,\ldots,P_r}, with degree and coefficients bounded by {C_{D,d,r}}.

I recently reposted my favourite logic puzzle, namely the blue-eyed islander puzzle. I am fond of this puzzle because in order to properly understand the correct solution (and to properly understand why the alternative solution is incorrect), one has to think very clearly (but unintuitively) about the nature of knowledge.

There is however an additional subtlety to the puzzle that was pointed out in comments, in that the correct solution to the puzzle has two components, a (necessary) upper bound and a (possible) lower bound (I’ll explain this further below the fold, in order to avoid blatantly spoiling the puzzle here). Only the upper bound is correctly explained in the puzzle (and even then, there are some slight inaccuracies, as will be discussed below). The lower bound, however, is substantially more difficult to establish, in part because the bound is merely possible and not necessary. Ultimately, this is because to demonstrate the upper bound, one merely has to show that a certain statement is logically deducible from an islander’s state of knowledge, which can be done by presenting an appropriate chain of logical deductions. But to demonstrate the lower bound, one needs to show that certain statements are not logically deducible from an islander’s state of knowledge, which is much harder, as one has to rule out all possible chains of deductive reasoning from arriving at this particular conclusion. In fact, to rigorously establish such impossiblity statements, one ends up having to leave the “syntactic” side of logic (deductive reasoning), and move instead to the dual “semantic” side of logic (creation of models). As we shall see, semantics requires substantially more mathematical setup than syntax, and the demonstration of the lower bound will therefore be much lengthier than that of the upper bound.

To complicate things further, the particular logic that is used in the blue-eyed islander puzzle is not the same as the logics that are commonly used in mathematics, namely propositional logic and first-order logic. Because the logical reasoning here depends so crucially on the concept of knowledge, one must work instead with an epistemic logic (or more precisely, an epistemic modal logic) which can properly work with, and model, the knowledge of various agents. To add even more complication, the role of time is also important (an islander may not know a certain fact on one day, but learn it on the next day), so one also needs to incorporate the language of temporal logic in order to fully model the situation. This makes both the syntax and semantics of the logic quite intricate; to see this, one only needs to contemplate the task of programming a computer with enough epistemic and temporal deductive reasoning powers that it would be able to solve the islander puzzle (or even a smaller version thereof, say with just three or four islanders) without being deliberately “fed” the solution. (The fact, therefore, that humans can grasp the correct solution without any formal logical training is therefore quite remarkable.)

As difficult as the syntax of temporal epistemic modal logic is, though, the semantics is more intricate still. For instance, it turns out that in order to completely model the epistemic state of a finite number of agents (such as 1000 islanders), one requires an infinite model, due to the existence of arbitrarily long nested chains of knowledge (e.g. “{A} knows that {B} knows that {C} knows that {D} has blue eyes”), which cannot be automatically reduced to shorter chains of knowledge. Furthermore, because each agent has only an incomplete knowledge of the world, one must take into account multiple hypothetical worlds, which differ from the real world but which are considered to be possible worlds by one or more agents, thus introducing modality into the logic. More subtly, one must also consider worlds which each agent knows to be impossible, but are not commonly known to be impossible, so that (for instance) one agent is willing to admit the possibility that another agent considers that world to be possible; it is the consideration of such worlds which is crucial to the resolution of the blue-eyed islander puzzle. And this is even before one adds the temporal aspect (e.g. “On Tuesday, {A} knows that on Monday, {B} knew that by Wednesday, {C} will know that {D} has blue eyes”).

Despite all this fearsome complexity, it is still possible to set up both the syntax and semantics of temporal epistemic modal logic in such a way that one can formulate the blue-eyed islander problem rigorously, and in such a way that one has both an upper and a lower bound in the solution. The purpose of this post is to construct such a setup and to explain the lower bound in particular. The same logic is also useful for analysing another well-known paradox, the unexpected hanging paradox, and I will do so at the end of the post. Note though that there is more than one way to set up epistemic logics, and they are not all equivalent to each other.

(On the other hand, for puzzles such as the islander puzzle in which there are only a finite number of atomic propositions and no free variables, one at least can avoid the need to admit predicate logic, in which one has to discuss quantifiers such as {\forall} and {\exists}. A fully formed predicate temporal epistemic modal logic would indeed be of terrifying complexity.)

Our approach here will be a little different from the approach commonly found in the epistemic logic literature, in which one jumps straight to “arbitrary-order epistemic logic” in which arbitrarily long nested chains of knowledge (“{A} knows that {B} knows that {C} knows that \ldots”) are allowed. Instead, we will adopt a hierarchical approach, recursively defining for {k=0,1,2,\ldots} a “{k^{th}}-order epistemic logic” in which knowledge chains of depth up to {k}, but no greater, are permitted. The arbitrarily order epistemic logic is then obtained as a limit (a direct limit on the syntactic side, and an inverse limit on the semantic side, which is dual to the syntactic side) of the finite order epistemic logics.

I should warn that this is going to be a rather formal and mathematical post. Readers who simply want to know the answer to the islander puzzle would probably be better off reading the discussion at the puzzle’s own blog post instead.

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One of the key difficulties in performing analysis in infinite-dimensional function spaces, as opposed to finite-dimensional vector spaces, is that the Bolzano-Weierstrass theorem no longer holds: a bounded sequence in an infinite-dimensional function space need not have any convergent subsequences (when viewed using the strong topology). To put it another way, the closed unit ball in an infinite-dimensional function space usually fails to be (sequentially) compact.

As compactness is such a useful property to have in analysis, various tools have been developed over the years to try to salvage some sort of substitute for the compactness property in infinite-dimensional spaces. One of these tools is concentration compactness, which was discussed previously on this blog. This can be viewed as a compromise between weak compactness (which is true in very general circumstances, but is often too weak for applications) and strong compactness (which would be very useful in applications, but is usually false), in which one obtains convergence in an intermediate sense that involves a group of symmetries acting on the function space in question.

Concentration compactness is usually stated and proved in the language of standard analysis: epsilons and deltas, limits and supremas, and so forth. In this post, I wanted to note that one could also state and prove the basic foundations of concentration compactness in the framework of nonstandard analysis, in which one now deals with infinitesimals and ultralimits instead of epsilons and ordinary limits. This is a fairly mild change of viewpoint, but I found it to be informative to view this subject from a slightly different perspective. The nonstandard proofs require a fair amount of general machinery to set up, but conversely, once all the machinery is up and running, the proofs become slightly shorter, and can exploit tools from (standard) infinitary analysis, such as orthogonal projections in Hilbert spaces, or the continuous-pure point decomposition of measures. Because of the substantial amount of setup required, nonstandard proofs tend to have significantly more net complexity than their standard counterparts when it comes to basic results (such as those presented in this post), but the gap between the two narrows when the results become more difficult, and for particularly intricate and deep results it can happen that nonstandard proofs end up being simpler overall than their standard analogues, particularly if the nonstandard proof is able to tap the power of some existing mature body of infinitary mathematics (e.g. ergodic theory, measure theory, Hilbert space theory, or topological group theory) which is difficult to directly access in the standard formulation of the argument.

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Many structures in mathematics are incomplete in one or more ways. For instance, the field of rationals {{\bf Q}} or the reals {{\bf R}} are algebraically incomplete, because there are some non-trivial algebraic equations (such as {x^2=2} in the case of the rationals, or {x^2=-1} in the case of the reals) which could potentially have solutions (because they do not imply a necessarily false statement, such as {1=0}, just using the laws of algebra), but do not actually have solutions in the specified field.

Similarly, the rationals {{\bf Q}}, when viewed now as a metric space rather than as a field, are also metrically incomplete, beause there exist sequences in the rationals (e.g. the decimal approximations {3, 3.1, 3.14, 3.141, \ldots} of the irrational number {\pi}) which could potentially converge to a limit (because they form a Cauchy sequence), but do not actually converge in the specified metric space.

A third type of incompleteness is that of logical incompleteness, which applies now to formal theories rather than to fields or metric spaces. For instance, Zermelo-Frankel-Choice (ZFC) set theory is logically incomplete, because there exist statements (such as the consistency of ZFC) which could potentially be provable by the theory (because it does not lead to a contradiction, or at least so we believe, just from the axioms and deductive rules of the theory), but is not actually provable in this theory.

A fourth type of incompleteness, which is slightly less well known than the above three, is what I will call elementary incompleteness (and which model theorists call the failure of the countable saturation property). It applies to any structure that is describable by a first-order language, such as a field, a metric space, or a universe of sets. For instance, in the language of ordered real fields, the real line {{\bf R}} is elementarily incomplete, because there exists a sequence of statements (such as the statements {0 < x < 1/n} for natural numbers {n=1,2,\ldots}) in this language which are potentially simultaneously satisfiable (in the sense that any finite number of these statements can be satisfied by some real number {x}) but are not actually simultaneously satisfiable in this theory.

In each of these cases, though, it is possible to start with an incomplete structure and complete it to a much larger structure to eliminate the incompleteness. For instance, starting with an arbitrary field {k}, one can take its algebraic completion (or algebraic closure) {\overline{k}}; for instance, {{\bf C} = \overline{{\bf R}}} can be viewed as the algebraic completion of {{\bf R}}. This field is usually significantly larger than the original field {k}, but contains {k} as a subfield, and every element of {\overline{k}} can be described as the solution to some polynomial equation with coefficients in {k}. Furthermore, {\overline{k}} is now algebraically complete (or algebraically closed): every polynomial equation in {\overline{k}} which is potentially satisfiable (in the sense that it does not lead to a contradiction such as {1=0} from the laws of algebra), is actually satisfiable in {\overline{k}}.

Similarly, starting with an arbitrary metric space {X}, one can take its metric completion {\overline{X}}; for instance, {{\bf R} = \overline{{\bf Q}}} can be viewed as the metric completion of {{\bf Q}}. Again, the completion {\overline{X}} is usually much larger than the original metric space {X}, but contains {X} as a subspace, and every element of {\overline{X}} can be described as the limit of some Cauchy sequence in {X}. Furthermore, {\overline{X}} is now a complete metric space: every sequence in {\overline{X}} which is potentially convergent (in the sense of being a Cauchy sequence), is now actually convegent in {\overline{X}}.

In a similar vein, we have the Gödel completeness theorem, which implies (among other things) that for any consistent first-order theory {T} for a first-order language {L}, there exists at least one completion {\overline{T}} of that theory {T}, which is a consistent theory in which every sentence in {L} which is potentially true in {\overline{T}} (because it does not lead to a contradiction in {\overline{T}}) is actually true in {\overline{T}}. Indeed, the completeness theorem provides at least one model (or structure) {{\mathfrak U}} of the consistent theory {T}, and then the completion {\overline{T} = \hbox{Th}({\mathfrak U})} can be formed by interpreting every sentence in {L} using {{\mathfrak U}} to determine its truth value. Note, in contrast to the previous two examples, that the completion is usually not unique in any way; a theory {T} can have multiple inequivalent models {{\mathfrak U}}, giving rise to distinct completions of the same theory.

Finally, if one starts with an arbitrary structure {{\mathfrak U}}, one can form an elementary completion {{}^* {\mathfrak U}} of it, which is a significantly larger structure which contains {{\mathfrak U}} as a substructure, and such that every element of {{}^* {\mathfrak U}} is an elementary limit of a sequence of elements in {{\mathfrak U}} (I will define this term shortly). Furthermore, {{}^* {\mathfrak U}} is elementarily complete; any sequence of statements that are potentially simultaneously satisfiable in {{}^* {\mathfrak U}} (in the sense that any finite number of statements in this collection are simultaneously satisfiable), will actually be simultaneously satisfiable. As we shall see, one can form such an elementary completion by taking an ultrapower of the original structure {{\mathfrak U}}. If {{\mathfrak U}} is the standard universe of all the standard objects one considers in mathematics, then its elementary completion {{}^* {\mathfrak U}} is known as the nonstandard universe, and is the setting for nonstandard analysis.

As mentioned earlier, completion tends to make a space much larger and more complicated. If one algebraically completes a finite field, for instance, one necessarily obtains an infinite field as a consequence. If one metrically completes a countable metric space with no isolated points, such as {{\bf Q}}, then one necessarily obtains an uncountable metric space (thanks to the Baire category theorem). If one takes a logical completion of a consistent first-order theory that can model true arithmetic, then this completion is no longer describable by a recursively enumerable schema of axioms, thanks to Gödel’s incompleteness theorem. And if one takes the elementary completion of a countable structure, such as the integers {{\bf Z}}, then the resulting completion {{}^* {\bf Z}} will necessarily be uncountable.

However, there are substantial benefits to working in the completed structure which can make it well worth the massive increase in size. For instance, by working in the algebraic completion of a field, one gains access to the full power of algebraic geometry. By working in the metric completion of a metric space, one gains access to powerful tools of real analysis, such as the Baire category theorem, the Heine-Borel theorem, and (in the case of Euclidean completions) the Bolzano-Weierstrass theorem. By working in a logically and elementarily completed theory (aka a saturated model) of a first-order theory, one gains access to the branch of model theory known as definability theory, which allows one to analyse the structure of definable sets in much the same way that algebraic geometry allows one to analyse the structure of algebraic sets. Finally, when working in an elementary completion of a structure, one gains a sequential compactness property, analogous to the Bolzano-Weierstrass theorem, which can be interpreted as the foundation for much of nonstandard analysis, as well as providing a unifying framework to describe various correspondence principles between finitary and infinitary mathematics.

In this post, I wish to expand upon these above points with regard to elementary completion, and to present nonstandard analysis as a completion of standard analysis in much the same way as, say, complex algebra is a completion of real algebra, or real metric geometry is a completion of rational metric geometry.

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This is the third in a series of posts on the “no self-defeating object” argument in mathematics – a powerful and useful argument based on formalising the observation that any object or structure that is so powerful that it can “defeat” even itself, cannot actually exist.   This argument is used to establish many basic impossibility results in mathematics, such as Gödel’s theorem that it is impossible for any sufficiently sophisticated formal axiom system to prove its own consistency, Turing’s theorem that it is impossible for any sufficiently sophisticated programming language to solve its own halting problem, or Cantor’s theorem that it is impossible for any set to enumerate its own power set (and as a corollary, the natural numbers cannot enumerate the real numbers).

As remarked in the previous posts, many people who encounter these theorems can feel uneasy about their conclusions, and their method of proof; this seems to be particularly the case with regard to Cantor’s result that the reals are uncountable.   In the previous post in this series, I focused on one particular aspect of the standard proofs which one might be uncomfortable with, namely their counterfactual nature, and observed that many of these proofs can be largely (though not completely) converted to non-counterfactual form.  However, this does not fully dispel the sense that the conclusions of these theorems – that the reals are not countable, that the class of all sets is not itself a set, that truth cannot be captured by a predicate, that consistency is not provable, etc. – are highly unintuitive, and even objectionable to “common sense” in some cases.

How can intuition lead one to doubt the conclusions of these mathematical results?  I believe that one reason is because these results are sensitive to the amount of vagueness in one’s mental model of mathematics.  In the formal mathematical world, where every statement is either absolutely true or absolutely false with no middle ground, and all concepts require a precise definition (or at least a precise axiomatisation) before they can be used, then one can rigorously state and prove Cantor’s theorem, Gödel’s theorem, and all the other results mentioned in the previous posts without difficulty.  However, in the vague and fuzzy world of mathematical intuition, in which one’s impression of the truth or falsity of a statement may be influenced by recent mental reference points, definitions are malleable and blurry with no sharp dividing lines between what is and what is not covered by such definitions, and key mathematical objects may be incompletely specified and thus “moving targets” subject to interpretation, then one can argue with some degree of justification that the conclusions of the above results are incorrect; in the vague world, it seems quite plausible that one can always enumerate all the real numbers “that one needs to”, one can always justify the consistency of one’s reasoning system, one can reason using truth as if it were a predicate, and so forth.    The impossibility results only kick in once one tries to clear away the fog of vagueness and nail down all the definitions and mathematical statements precisely.  (To put it another way, the no-self-defeating object argument relies very much on the disconnected, definite, and absolute nature of the boolean truth space \{\hbox{true},\hbox{ false}\} in the rigorous mathematical world.)

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One notable feature of mathematical reasoning is the reliance on counterfactual thinking – taking a hypothesis (or set of hypotheses) which may or may not be true, and following it (or them) to its logical conclusion.  For instance, most propositions in mathematics start with a set of hypotheses (e.g. “Let n be a natural number such that …”), which may or may not apply to the particular value of n one may have in mind.  Or, if one ever argues by dividing into separate cases (e.g. “Case 1: n is even. … Case 2: n is odd.  …”), then for any given n, at most one of these cases would actually be applicable, with the other cases being counterfactual alternatives.     But the purest example of counterfactual thinking in mathematics comes when one employs a proof by contradiction (or reductio ad absurdum) – one introduces a hypothesis that in fact has no chance of being true at all (e.g. “Suppose for sake of contradiction that \sqrt{2} is equal to the ratio p/q of two natural numbers.”), and proceeds to demonstrate this fact by showing that this hypothesis leads to absurdity.

Experienced mathematicians are so used to this type of counterfactual thinking that it is sometimes difficult for them to realise that it this type of thinking is not automatically intuitive for students or non-mathematicians, who can anchor their thinking on the single, “real” world to the extent that they cannot easily consider hypothetical alternatives.  This can lead to confused exchanges such as the following:

Lecturer: “Theorem.  Let p be a prime number.  Then…”

Student: “But how do you know that p is a prime number?  Couldn’t it be composite?”


Lecturer: “Now we see what the function f does when we give it the input of x+dx instead.  …”

Student: “But didn’t you just say that the input was equal to x just a moment ago?”

This is not to say that counterfactual thinking is not encountered at all outside of mathematics.  For instance, an obvious source of counterfactual thinking occurs in fictional writing or film, particularly in speculative fiction such as science fiction, fantasy, or alternate history.  Here, one can certainly take one or more counterfactual hypotheses (e.g. “what if magic really existed?”) and follow them to see what conclusions would result.  The analogy between this and mathematical counterfactual reasoning is not perfect, of course: in fiction, consequences are usually not logically entailed by their premises, but are instead driven by more contingent considerations, such as the need to advance the plot, to entertain or emotionally affect the reader, or to make some moral or ideological point, and these types of narrative elements are almost completely absent in mathematical writing.  Nevertheless, the analogy can be somewhat helpful when one is first coming to terms with mathematical reasoning.  For instance, the mathematical concept of a proof by contradiction can be viewed as roughly analogous in some ways to such literary concepts as satire, dark humour, or absurdist fiction, in which one takes a premise specifically with the intent to derive absurd consequences from it.  And if the proof of (say) a lemma is analogous to a short story, then the statement of that lemma can be viewed as analogous to the moral of that story.

Another source of counterfactual thinking outside of mathematics comes from simulation, when one feeds some initial data or hypotheses (that may or may not correspond to what actually happens in the real world) into a simulated environment (e.g. a piece of computer software, a laboratory experiment, or even just a thought-experiment), and then runs the simulation to see what consequences result from these hypotheses.   Here, proof by contradiction is roughly analogous to the “garbage in, garbage out” phenomenon that is familiar to anyone who has worked with computers: if one’s initial inputs to a simulation are not consistent with the hypotheses of that simulation, or with each other, one can obtain bizarrely illogical (and sometimes unintentionally amusing) outputs as a result; and conversely, such outputs can be used to detect and diagnose problems with the data, hypotheses, or implementation of the simulation.

Despite the presence of these non-mathematical analogies, though, proofs by contradiction are still often viewed with suspicion and unease by many students of mathematics.  Perhaps the quintessential example of this is the standard proof of Cantor’s theorem that the set {\bf R} of real numbers is uncountable.  This is about as short and as elegant a proof by contradiction as one can have without being utterly trivial, and despite this (or perhaps because of this) it seems to offend the reason of many people when they are first exposed to it, to an extent far greater than most other results in mathematics.  (The only other two examples I know of that come close to doing this are the fact that the real number 0.999\ldots is equal to 1, and the solution to the blue-eyed islanders puzzle.)

Some time ago on this blog, I collected a family of well-known results in mathematics that were proven by contradiction, and specifically by a type of argument that I called the “no self-defeating object” argument; that any object that was so ridiculously overpowered that it could be used to “defeat” its own existence, could not actually exist.  Many basic results in mathematics can be phrased in this manner: not only Cantor’s theorem, but Euclid’s theorem on the infinitude of primes, Gödel’s incompleteness theorem, or the conclusion (from Russell’s paradox) that the class of all sets cannot itself be a set.

I presented each of these arguments in the usual “proof by contradiction” manner; I made the counterfactual hypothesis that the impossibly overpowered object existed, and then used this to eventually derive a contradiction.  Mathematically, there is nothing wrong with this reasoning, but because the argument spends almost its entire duration inside the bizarre counterfactual universe caused by an impossible hypothesis, readers who are not experienced with counterfactual thinking may view these arguments with unease.

It was pointed out to me, though (originally with regards to Euclid’s theorem, but the same point in fact applies to the other results I presented) that one can pull a large fraction of each argument out of this counterfactual world, so that one can see most of the argument directly, without the need for any intrinsically impossible hypotheses.  This is done by converting the “no self-defeating object” argument into a logically equivalent “any object can be defeated” argument, with the former then being viewed as an immediate corollary of the latter.  This change is almost trivial to enact (it is often little more than just taking the contrapositive of the original statement), but it does offer a slightly different “non-counterfactual” (or more precisely, “not necessarily counterfactual”) perspective on these arguments which may assist in understanding how they work.

For instance, consider the very first no-self-defeating result presented in the previous post:

Proposition 1 (No largest natural number). There does not exist a natural number N that is larger than all the other natural numbers.

This is formulated in the “no self-defeating object” formulation.  But it has a logically equivalent “any object can be defeated” form:

Proposition 1′. Given any natural number N, one can find another natural number N' which is larger than N.

Proof. Take N' := N+1. \Box

While Proposition 1 and Proposition 1′ are logically equivalent to each other, note one key difference: Proposition 1′ can be illustrated with examples (e.g. take N = 100, so that the proof gives N'=101 ), whilst Proposition 1 cannot (since there is, after all, no such thing as a largest natural number).  So there is a sense in which Proposition 1′ is more “non-counterfactual” or  “constructive” than the “counterfactual” Proposition 1.

In a similar spirit, Euclid’s theorem (which we give using the numbering from the previous post),

Proposition 3. There are infinitely many primes.

can be recast in “all objects can be defeated” form as

Proposition 3′.  Let p_1,\ldots,p_n be a collection of primes.   Then there exists a prime q which is distinct from any of the primes p_1,\ldots,p_n.

Proof. Take q to be any prime factor of p_1 \ldots p_n + 1 (for instance, one could take the smallest prime factor, if one wished to be completely concrete).   Since p_1 \ldots p_n + 1 is not divisible by any of the primes p_1,\ldots,p_n, q must be distinct from all of these primes.  \Box

One could argue that  there was a slight use of proof by contradiction in the proof of Proposition 3′ (because one had to briefly entertain and then rule out the counterfactual possibility that q was equal to one of the p_1,\ldots,p_n), but the proposition itself is not inherently counterfactual, as  it does not make as patently impossible a hypothesis as a finite enumeration of the primes.  Incidentally, it can be argued that the proof of Proposition 3′ is closer in spirit to Euclid’s original proof of his theorem, than the proof of Proposition 3 that is usually given today.  Again, Proposition 3′ is “constructive”; one can apply it to any finite list of primes, say 2, 3, 5, and it will actually exhibit a prime not in that list (in this case, 31).  The same cannot be said of Proposition 3, despite the logical equivalence of the two statements.

[Note: the article below may make more sense if one first reviews the previous blog post on the "no self-defeating object".  For instance, the section and theorem numbering here is deliberately chosen to match that of the preceding post.]

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