This week, Henry Towsner concluded his portion of reading seminar of the Hrushovski paper, by discussing (a weaker, simplified version of) main model-theoretic theorem (Theorem 3.4 of Hrushovski), and described how this theorem implied the combinatorial application in Corollary 1.2 of Hrushovski. The presentation here differs slightly from that in Hrushovski’s paper, for instance by avoiding mention of the more general notions of S1 ideals and forking.

Here is a collection of resources so far on the Hrushovski paper:

• Henry Towsner’s notes (which most of Notes 2-4 have been based on);
• Alex Usvyatsov’s notes on the derivation of Corollary 1.2 (broadly parallel to the notes here);
• Lou van den Dries’ notes (covering most of what we have done so far, and also material on stable theories); and
• Anand Pillay’s sketch of a simplified proof of Theorem 1.1.

— 1. Theorem 3.4 —

Here is a weakened version of Hrushovski’s Theorem 3.4:

Theorem 1 Let be a countable model of a language extending the language of groups, with a universal extension . Let be a continuous, invariant Keisler measure that is also invariant under translations. Let be a symmetric definable subset of with and finite, and let be the group generated by . Let be a wide type over contained in , and suppose that for every , there exists such that and are both wide.

Then the set is a normal subgroup of .

Remark 1 The condition about and seems to be a statement that there exist plenty of pairs in that are in “general position” somehow. In the case of abelian groups, it seems that this hypothesis not necessary. The hypothesis is for all , but by homogeneity it suffices to verify it for a single (since the action of the automorphism group of is transitive on ).

Remark 2 Interestingly, it seems that no doubling condition on is needed to prove this theorem, but the doubling condition arises when one wants to put to use.

Remark 3 As is wide, is wide also. If we also assume that every finite product has finite measure, this leads to the consequence that the index of in does not exceed in cardinality (i.e. has bounded index, see Lemma 1 of Notes 2). Indeed, if this were not the case, then one could find more than disjoint cosets in , and hence in some finite product . By Section 4 of Notes 2, is -definable, and so is the intersection of countably many definable sets . For any distinct , are disjoint, thus by saturation we have non-empty for some integer . By the Erdös-Rado theorem (an infinitary analogue of Ramsey’s theorem), we can then ass to a countable set of indices in which the are constant, thus we have a countable set of disjoint translates , which live in some finite product . But as is wide, has positive measure, while has finite leasure, leading to a contradiction.

Remark 4 The set can also be defined in a more “characteristic” fashion as the smallest -definable subgroup of of bounded index; we’ll return to this point in a later note (I think).

Remark 5 Here is one example of how the group emerges. Start with a finite combinatorial model, in which is a discrete interval and the group is the integers; we normalise to have measure . Taking ultralimits, we end up with a non-standard discrete interval in the non-standard integers, with being the counting measure normalised to give a measure of . (This is not yet a saturated model, but yet us ignore this issue.) Note that the subsets are definable for any standard natural number . Their intersection is a wide subgroup; if was a “generic” type in then would be wide, and would generate (this is vaguely reminiscent of the “Bogulybov theorem” in additive combinatorics).

To apply Theorem 1, we need the following auxiliary lemma (a weakened version of Hrushovski Lemma 2.15):

Lemma 2 Let be a definable set of positive measure. Then we can refine to a wide type such that for every , there exists such that and are both wide.

This lemma can be viewed as a more technical variant of Lemma 2 from Notes 2.

To prove this lemma we need a technical sublemma:

Lemma 3 Let be a definable set of positive measure, and suppose that the product is covered by two definable sets (these are sets of pairs, of course). Then one can refine to a definable subset of positive measure, such that at least one of the following holds:

• (i) For every , the set has positive measure; or
• (ii) For every , the set has positive measure.

Proof: We can refine to be contained in . At least one of must have positive measure; let’s say that . We can find a definable set between and . If any of these has positive measure, then we can just take , so we may assume instead that for all , which implies that by Fubini’s theorem (which can be extended to Kiesler measures, at least in the weak form we need here), giving the desired contradiction.

Now we prove Lemma 2. By starting with and refining repeatedly using Lemma 3 (enumerating all the definable sets, and pairs of definable sets , so that each pair is revisited infinitely often) one can find a type refining with the property that whenever is a definable set containing , and is covered by two definable sets , then either has positive measure for all , or has positive measure for all . In particular (setting and empty) this implies that is wide.

Now let , and define to be the type , refined using the complement of all the sets of the form for definable with , and the complement of the sets of the form for definable with (note the asymmetry here!). We claim that this collection of sentences is still finitely satisfiable. Indeed, if this were not the case, would have to be covered by for some definable with . Write , then the set is definable and contains , and thus contains . If we set , we thus see that is a definable set containing and , contradicting the construction of .

As is finitely satisfiable, we may extend it to a type over . Now let be any realisation of . We claim that is wide. For if not, then would be contained in a set of measure zero for some definable , but by construction and hence is contained in the complement of , a contradiction.

We similarly claim that is wide. For if not, then is contained in a set of measure zero for some definable . since and have the same type over , has measure zero also, and so is contained in the complement of by construction. In particular, lies outside , which is inconsistent with lying in . This proves Lemma 2.

Remark 6 As I understand it, the argument simplifies in the stable case, when the wideness of over is equivalent to the wideness of over .

— 2. Corollary 1.2 —

Assuming Theorem 1, we now establish the following combinatorial consequence, which is (a slightly weaker version of) Corollary 1.2 of Hrushovski:

Corollary 4 Let be positive integers. Then for sufficiently close to , the following statement is true: whenever is a finite subset of a group with the bounded quintipling property , where , and for a proportion at least of the -tuples , one has (recall that ). Then there exists a subgroup of such that is contained in at most cosets of .

We have weakened things a bit here by drawing the from rather than (and also weakening bounded tripling to bounded quintipling); this is in order to achieve some minor simplifications in the proof.

We now prove the corollary, by the usual “compactness and contradiction” method. If the corollary failed for some , then one can find a sequence of groups with subsets of tripling uniformly bounded by , such that the proportion of -tuples in with approaches , but such that there is no subgroup of which can cover by cosets.

Taking an ultralimit (and using counting measure normalised so that has mass ), we obtain a subset of a group and a continuous, invariant, translation-invariant Keisler measure so that , , and such that the set (say) has measure . (Strictly speaking, one has to replace by a definable predicate between and , but let’s ignore this technicality.) We then extend to a universal model .

The set has positive measure (at least , in fact), so by Lemma 2, we may refine it to a wide type obeying the hypotheses of Theorem 1, so that the set is a group. Note that this set is contained in .

Since contains the wide type , it is itself wide. By Section 4 of Notes 2, is -definable, and is thus the intersection of definable sets , which have positive measure as is wide; we can place this inside the definable set . As has full measure in , we see that intersects for every , by saturation, must also intersect . In particular, there exists in such that

On the other hand, as is normal in , the set lies in . This implies that we cannot find more than disjoint translates , . As a consequence, is covered by at most translates of , and so can be covered by such translates, including itself. In particular, is both -definable and -definable (it is the complement of all the other cosets in ). But any set which is both -definable and -definable has to be definable. (Proof: we can write as the intersection of definable sets and also as the union of definable sets . By saturation, must be empty for at least one , and the claim follows.)

To summarise, is now a definable group which can cover by at most cosets. This is a first-order statement and so descends back to the original finitary models, but this then contradicts the construction of such sets, and proves the Corollary.

Remark 7 The argument not only shows the existence of the set , but shows that this set is “definable”, in the sense that there are a finite number of first-order expressions using and normalised cardinality, one of which is guaranteed to generate whenever the hypotheses of Corollary 4 are satisfied. Indeed, if this were not the case, one could construct a sequence of counterexamples for which no such recipe for constructing would work for sufficiently late elements of this sequence, and then run the above argument to obtain a contradiction.

A fundamental tool in any mathematician’s toolkit is that of reductio ad absurdum: showing that a statement is false by assuming first that is true, and showing that this leads to a logical contradiction. A particulary pure example of reductio ad absurdum occurs when establishing the non-existence of a hypothetically overpowered object or structure , by showing that ’s powers are “self-defeating”: the very existence of and its powers can be used (by some clever trick) to construct a counterexample to that power. Perhaps the most well-known example of a self-defeating object comes from the omnipotence paradox in philosophy (“Can an omnipotent being create a rock so heavy that He cannot lift it?”); more generally, a large number of other paradoxes in logic or philosophy can be reinterpreted as a proof that a certain overpowered object or structure does not exist.

In mathematics, perhaps the first example of a self-defeating object one encounters is that of a largest natural number:

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

Proof: Suppose for contradiction that there was such a largest natural number . Then is also a natural number which is strictly larger than , contradicting the hypothesis that is the largest natural number.

Note the argument does not apply to the extended natural number system in which one adjoins an additional object beyond the natural numbers, because is defined equal to . However, the above argument does show that the existence of a largest number is not compatible with the Peano axioms.

This argument, by the way, is perhaps the only mathematical argument I know of which is routinely taught to primary school children by other primary school children, thanks to the schoolyard game of naming the largest number. It is arguably one’s first exposure to a mathematical non-existence result, which seems innocuous at first but can be surprisingly deep, as such results preclude in advance all future attempts to establish existence of that object, no matter how much effort or ingenuity is poured into this task. One sees this in a typical instance of the above schoolyard game; one player tries furiously to cleverly construct some impressively huge number , but no matter how much effort is expended in doing so, the player is defeated by the simple response “… plus one!” (unless, of course, is infinite, ill-defined, or otherwise not a natural number).

It is not only individual objects (such as natural numbers) which can be self-defeating; structures (such as orderings or enumerations) can also be self-defeating. (In modern set theory, one considers structures to themselves be a kind of object, and so the distinction between the two concepts is often blurred.) Here is one example (related to, but subtly different from, the previous one):

Proposition 2 (The natural numbers cannot be finitely enumerated) The natural numbers cannot be written as for any finite collection of natural numbers.

Proof: Suppose for contradiction that such an enumeration existed. Then consider the number ; this is a natural number, but is larger than (and hence not equal to) any of the natural numbers , contradicting the hypothesis that is enumerated by .

Here it is the enumeration which is self-defeating, rather than any individual natural number such as or . (For this post, we allow enumerations to contain repetitions.)

The above argument may seem trivial, but a slight modification of it already gives an important result, namely Euclid’s theorem:

Proposition 3 (The primes cannot be finitely enumerated) The prime numbers cannot be written as for any finite collection of prime numbers.

Proof: Suppose for contradiction that such an enumeration existed. Then consider the natural number ; this is a natural number larger than which is not divisible by any of the primes . But, by the fundamental theorem of arithmetic (or by the method of Infinite descent, which is another classic application of reductio ad absurdum), every natural number larger than must be divisible by some prime, contradicting the hypothesis that is enumerated by .

Remark 1 Continuing the number-theoretic theme, the “dueling conspiracies” arguments in a previous blog post can also be viewed as an instance of this type of “no-self-defeating-object”; for instance, a zero of the Riemann zeta function at implies that the primes anti-correlate almost completely with the multiplicative function , which is self-defeating because it also implies complete anti-correlation with , and the two are incompatible. Thus we see that the prime number theorem (a much stronger version of Proposition 3) also emerges from this type of argument.

In this post I would like to collect several other well-known examples of this type of “no self-defeating object” argument. Each of these is well studied, and probably quite familiar to many of you, but I feel that by collecting them all in one place, the commonality of theme between these arguments becomes more apparent. (For instance, as we shall see, many well-known “paradoxes” in logic and philosophy can be interpreted mathematically as a rigorous “no self-defeating object” argument.)

— 1. Set theory —

Many famous foundational results in set theory come from a “no self-defeating object” argument. (Here, we shall be implicitly be using a standard axiomatic framework for set theory, such as Zermelo-Frankel set theory.) The basic idea here is that any sufficiently overpowered set-theoretic object is capable of encoding a version of the liar paradox (“this sentence is false”, or more generally a statement which can be shown to be logically equivalent to its negation) and thus lead to a contradiction. Consider for instance this variant of Russell’s paradox:

Proposition 4 (No universal set) There does not exist a set which contains all sets (including itself).

Proof: Suppose for contradiction that there existed a universal set which contained all sets. Using the axiom schema of specification, one can then construct the set

of all sets in the universe which did not contain themselves. As is universal, is contained in . But then, by definition of , one sees that if and only if , a contradiction.

Remark 2 As a corollary, there also does not exist any set which contains all other sets (not including itself), because the set would then be universal.

One can “localise” the above argument to a smaller domain than the entire universe, leading to the important

Proposition 5 (Cantor’s theorem) Let be a set. Then the power set of cannot be enumerated by , i.e. one cannot write for some collection of subsets of .

Proof: Suppose for contradiction that there existed a set whose power set could be enumerated as for some . Using the axiom schema of specification, one can then construct the set

The set is an element of the power set . As is enumerated by , we have for some . But then by the definition of , one sees that if and only if , a contradiction.

As is well-known, one can adapt Cantor’s argument to the real line, showing that the reals are uncountable:

Proposition 6 (The real numbers cannot be countably enumerated) The real numbers cannot be written as for any countable collection of real numbers.

Proof: Suppose for contradiction that there existed a countable enumeration of by a sequence of real numbers. Consider the decimal expansion of each of these numbers. Note that, due to the well-known “” issue, the decimal expansion may be non-unique, but each real number has at most two decimal expansions. For each , let be a digit which is not equal to the digit of any of the decimal expansions of ; this is always possible because there are ten digits to choose from and at most two decimal expansions of . (One can avoid any implicit invocation of the axiom of choice here by setting to be (say) the least digit which is not equal to the digit of any of the decimal expansions of .) Then the real number given by the decimal expansion differs in the digit from any of the decimal expansions of for each , and so is distinct from every , a contradiction.

Remark 3 One can of course deduce Proposition 6 directly from Proposition 5, by using the decimal representation to embed into . But notice how the two arguments have a slightly different (though closely related) basis; the former argument proceeds by encoding the liar paradox, while the second proceeds by exploiting Cantor’s diagonal argument. The two perspectives are indeed a little different: for instance, Cantor’s diagonal argument can also be modified to establish the Arzela-Ascolí theorem, whereas I do not see any obvious way to prove that theorem by encoding the liar paradox.

Remark 4 It is an interesting psychological phenomenon that Proposition 6 is often considered far more unintuitive than any of the other propositions here, despite being in the same family of arguments; most people have no objection to the fact that every effort to finitely enumerate the natural numbers, for instance, is doomed to failure, but for some reason it is much harder to let go of the belief that, at some point, some sufficiently ingenious person will work out a way to countably enumerate the real numbers. I am not exactly sure why this disparity exists, but I suspect it is at least partly due to the fact that the rigorous construction of the real numbers is quite sophisticated and often not presented properly until the advanced undergraduate level. (Or perhaps it is because we do not play the game “enumerate the real numbers” often enough in schoolyards.)

Remark 5 One can also use the diagonal argument to show that any reasonable notion of a “constructible real number” must itself be non-constructive (this is related to the interesting number paradox). Part of the problem is that the question of determining whether a proposed construction of a real number is actually well-defined is a variant of the halting problem, which we will discuss below.

While a genuinely universal set is not possible in standard set theory, one at least has the notion of an ordinal, which contains all the ordinals less than it. (In the discussion below, we assume familiarity with the theory of ordinals.) One can modify the above arguments concerning sets to give analogous results about the ordinals. For instance:

Proposition 7 (Ordinals do not form a set) There does not exist a set that contains all the ordinals.

Proof: Suppose for contradiction that such a set existed. By the axiom schema of specification, one can then find a set which consists precisely of all the ordinals and nothing else. But then is an ordinal which is not contained in (by the axiom of foundation, also known as the axiom of regularity), a contradiction.

Remark 6 This proposition(together with the theory of ordinals) can be used to give a quick proof of Zorn’s lemma: see these lecture notes of mine for further discussion. Notice the similarity between this argument and the proof of Proposition 1.

Remark 7 Once one has Zorn’s lemma, one can show that various other classes of mathematical objects do not form sets. Consider for instance the class of all vector spaces. Observe that any chain of (real) vector spaces (ordered by inclusion) has an upper bound (namely the union or limit of these spaces); thus, if the class of all vector spaces was a set, then Zorn’s lemma would imply the existence of a maximal vector space . But one can simply adjoin an additional element not already in (e.g. ) to , and contradict this maximality. (An alternate proof: every object is an element of some vector space, and in particular every set is an element of some vector space. If the class of all vector spaces formed a set, then by the axiom of union, we see that union of all vector spaces is a set also, contradicting Proposition 4.)

One can localise the above argument to smaller cardinalities, for instance:

Proposition 8 (Countable ordinals are uncountable) There does not exist a countable enumeration of the countable ordinals. (Here we consider finite sets and countably infinite sets to both be countable.)

Proof: Suppose for contradiction that there exists a countable enumeration of the countable ordinals. Then the set is also a countable ordinal, as is the set . But is not equal to any of the (by the axiom of foundation), a contradiction.

Remark 8 With the axiom of choice, one can show the existence of uncountable ordinals (e.g. by well-ordering the reals), and then there exists a least uncountable ordinal . By construction, this ordinal consists precisely of all the countable ordinals, but is itself uncountable, much as consists precisely of all the finite natural numbers, but is itself infinite (Proposition 2). The least uncountable ordinal is notorious, among other things, for providing a host of counterexamples to various intuitively plausible assertions in point set topology, and in particular in showing that the topology of sufficiently uncountable spaces cannot always be adequately explored by countable objects such as sequences.

Remark 9 The existence of the least uncountable ordinal can explain why one cannot contradict Cantor’s theorem on the uncountability of the reals simply by iterating the diagonal argument (or any other algorithm) in an attempt to “exhaust” the reals. From transfinite induction we see that the diagonal argument allows one to assign a different real number to each countable ordinal, but this does not establish countability of the reals, because the set of all countable ordinals is itself uncountable. (This is similar to how one cannot contradict Proposition 4 by iterating the map, as the set of all finite natural numbers is itself infinite.) In any event, even once one reaches the first uncountable ordinal, one may not yet have completely exhausted the reals; for instance, using the diagonal argument given in the proof of Proposition 6, only the real numbers in the interval will ever be enumerated by this procedure. (Also, the question of whether all real numbers in can be enumerated by the iterated diagonal algorithm requires the continuum hypothesis, and even with this hypothesis I am not sure whether the statement is decidable.)

— 2. Logic —

The “no self-defeating object” argument leads to a number of important non-existence results in logic. Again, the basic idea is to show that a sufficiently overpowered logical structure will eventually lead to the existence of a self-contradictory statement, such as the liar paradox. To state examples of this properly, one unfortunately has to invest a fair amount of time in first carefully setting up the language and theory of logic. I will not do so here, and instead use informal English sentences as a proxy for precise logical statements to convey a taste (but not a completely rigorous description) of these logical results here.

The liar paradox itself – the inability to assign a consistent truth value to “this sentence is false” – can be viewed as an argument demonstrating that there is no consistent way to interpret (i.e. assign a truth value to) sentences, when the sentences are (a) allowed to be self-referential, and (b) allowed to invoke the very notion of truth given by this interpretation. One’s first impulse is to say that the difficulty here lies more with (a) than with (b), but there is a clever trick, known as Quining (or indirect self-reference), which allows one to modify the liar paradox to produce a non-self-referential statement to which one still cannot assign a consistent truth value. The idea is to work not with fully formed sentences , which have a single truth value, but instead with predicates , whose truth value depends on a variable in some range. For instance, may be “ is thirty-two characters long.”, and the range of may be the set of strings (i.e. finite sequences of characters); then for every string , the statement (formed by replacing every appearance of in with ) is either true or false. For instance, is true, but is false. Crucially, predicates are themselves strings, and can thus be fed into themselves as input; for instance, is false. If however is the predicate “ is sixty-five characters long.”, observe that is true.

Now consider the Quine predicate given by

“ is a predicate whose range is the set of strings, and is false.”

whose range is the set of strings. Thus, for any string , is the sentence

“ is a predicate whose range is the set of strings, and is false.”

This predicate is defined non-recursively, but the sentence captures the essence of the liar paradox: it is true if and only if it is false. This shows that there is no consistent way to interpret sentences in which the sentences are allowed to come from predicates, are allowed to use the concept of a string, and also allowed to use the concept of truth as given by that interpretation.

Note that the proof of Proposition 4 is basically the set-theoretic analogue of the above argument, with the connection being that one can identify a predicate with the set .

By making one small modification to the above argument – replacing the notion of truth with the related notion of provability – one obtains the celebrated Gödel’s (second) incompleteness theorem:

Theorem 9 (Gödel’s incompleteness theorem) (Informal statement) No consistent logical system which has the notion of a string, can provide a proof of its own logical consistency. (Note that a proof can be viewed as a certain type of string.)

Remark 10 Because one can encode strings in numerical form (e.g. using the ASCII code), it is also true (informally speaking) that no consistent logical system which has the notion of a natural number, can provide a proof of its own logical consistency.

Proof: (Informal sketch only) Suppose for contradiction that one had a consistent logical system inside of which its consistency could be proven. Now let be the predicate given by

“ is a predicate whose range is the set of strings, and is not provable”

and whose range is the set of strings. Define the Gödel sentence to be the string . Then is logically equivalent to the assertion “ is not provable”. Thus, if were false, then would be provable, which (by the consistency of the system) implies that is true, a contradiction; thus, must be true. Because the system is provably consistent, the above argument can be placed inside the system itself, to prove inside that system that must be true; thus is provable and is then false, a contradiction. (It becomes quite necessary to carefully distinguish the notions of truth and provability (both inside a system and externally to that system) in order to get this argument straight!)

Remark 11 It is not hard to show that a consistent logical system which can model the standard natural numbers cannot disprove its own consistency either (i.e. it cannot establish the statement that one can deduce a contradiction from the axioms in the systems in steps for some natural number ); thus the consistency of such a system is undecidable within that system. Thus this theorem strengthens the (more well known) first Gödel incompleteness theory, which asserts the existence of undecidable statements inside a consistent logical system which contains the concept of a string (or a natural number). On the other hand, the incompleteness theorem does not preclude the possibility that the consistency of a weak theory could be proven in a strictly stronger theory (e.g. the consistency of Peano arithmetic is provable in Zermelo-Frankel set theory).

Remark 12 One can use the incompleteness theorem to establish the undecidability of other overpowered problems. For instance, Matiyasevich’s theorem demonstrates that the problem of determining the solvability of a system of Diophantine equations is, in general, undecidable, because one can encode statements such as the consistency of set theory inside such a system.

— 3. Computability —

One can adapt these arguments in logic to analogous arguments in the theory of computation; the basic idea here is to show that a sufficiently overpowered computer program cannot exist, by feeding the source code for that program into the program itself (or some slight modification thereof) to create a contradiction. As with logic, a properly rigorous formalisation of the theory of computation would require a fair amount of preliminary machinery, for instance to define the concept of Turing machine (or some other universal computer), and so I will once again use informal English sentences as an informal substitute for a precise programming language.

A fundamental “no self-defeating object” argument in the subject, analogous to the other liar paradox type arguments encountered previously, is the Turing halting theorem:

Theorem 10 (Turing halting theorem) There does not exist a program which takes a string as input, and determines in finite time whether is a program (with no input) that halts in finite time.

Proof: Suppose for contradiction that such a program existed. Then one could easily modify to create a variant program , which takes a string as input, and halts if and only if is a program (with itself as input) that does not halts in finite time. Indeed, all has to do is call with the string , defined as the program (with no input) formed by declaring to be the input string for the program . If determines that does not halt, then halts; otherwise, if determines that does halt, then performs an infinite loop and does not halt. Then observe that will halt if and only if it does not halt, a contradiction.

Remark 13 As one can imagine from the proofs, this result is closely related to, but not quite identical with, the Gödel incompleteness theorem. That latter theorem implies that the question of whether a given program halts or not in general is undecidable (consider a program designed to search for proofs of the inconsistency of set theory). By contrast, the halting theorem (roughly speaking) shows that this question is uncomputable (i.e. there is no algorithm to decide halting in general) rather than undecidable (i.e. there are programs whose halting can neither be proven nor disproven).

On the other hand, the halting theorem can be used to establish the incompleteness theorem. Indeed, suppose that all statements in a suitably strong and consistent logical system were either provable or disprovable. Then one could build a program that determined whether an input string , when run as a program, halts in finite time, simply by searching for all proofs or disproofs of the statement “ halts in finite time”; this program is guaranteed to terminate with a correct answer by hypothesis.

Remark 14 While it is not possible for the halting problem for a given computing language to be computable in that language, it is certainly possible that it is computable in a strictly stronger language. When that is the case, one can then invoke Newcomb’s paradox to argue that the weaker language does not have unlimited “free will” in some sense.

Remark 15 In the language of recursion theory, the halting theorem asserts that the set of programs that do not halt is not a decidable set (or a recursive set). In fact, one can make the slightly stronger assertion that the set of programs that do not halt is not even a semi-decidable set (or a recursively enumerable set), i.e. there is no algorithm which takes a program as input and halts in finite time if and only if the input program does not halt. This is because the complementary set of programs that do halt is clearly semi-decidable (one simply runs the program until it halts, running forever if it does not), and so if the set of programs that do not halt is also semi-decidable, then it is decidable (by running both algorithms in parallel).

Remark 16 One can use the halting theorem to exclude overly general theories for certain types of mathematical objects. For instance, one cannot hope to find an algorithm to determine the existence of smooth solutions to arbitrary nonlinear partial differential equations, because it is possible to simulate a Turing machine using the laws of classical physics, which in turn can be modeled using (a moderately complicated system of) nonlinear PDE. Instead, progress in nonlinear PDE has instead proceeded by focusing on much more specific classes of such PDE (e.g. elliptic PDE, parabolic PDE, hyperbolic PDE, gauge theories, etc.).

One can place the halting theorem in a more “quantitative” form. Call a function computable if there exists a computer program which, when given a natural number as input, returns as output in finite time. Define the Busy Beaver function by setting to equal the largest output of any program of at most characters in length (and taking no input), which halts in finite time. Note that there are only finitely many such programs for any given , so is well-defined. On the other hand, it is uncomputable, even to upper bound:

Proposition 11 There does not exist a computable function such that one has for all .

Proof: Suppose for contradiction that there existed a computable function such that for all . We can use this to contradict the halting theorem, as follows. First observe that once the Busy Beaver function can be upper bounded by a computable function, then for any , the run time of any halting program of length at most can also be upper bounded by a computable function. This is because if a program of length halts in finite time, then a trivial modification of that program (of length larger than , but by a computable factor) is capable of outputting the run time of that program (by keeping track of a suitable “clock” variable, for instance). Applying the upper bound for Busy Beaver to that increased length, one obtains the bound on run time.

Now, to determine whether a given program halts in finite time or not, one simply simulates (runs) that program for time up to the computable upper bound of the possible running time of , given by the length of . If the program has not halted by then, then it never will. This provides a program obeying the hypotheses of the halting theorem, a contradiction.

Remark 17 A variant of the argument shows that grows faster than any computable function: thus if is computable, then for all sufficiently large . We leave the proof of this result as an exercise to the reader.

Remark 18 Sadly, the most important unsolved problem in complexity theory, namely the problem, does not seem to be susceptible to the no-self-defeating-object argument, basically because such arguments tend to be relativisable whereas the problem is not; see this earlier blog post for more discussion. On the other hand, one has the curious feature that many proposed proofs that appear to be self-defeating; this is most strikingly captured in the celebrated work of Razborov and Rudich, who showed (very roughly speaking) that any sufficiently “natural” proof that could be used to disprove the existence of an object closely related to the belief that , namely the existence of pseudorandom number generators. (I am told, though, that diagonalisation arguments can be used to prove some other inclusions or non-inclusions in complexity theory that are not subject to the relativisation barrier, though I do not know the details.)

— 4. Game theory —

Another basic example of the no-self-defeating-objects argument arises from game theory, namely the strategy stealing argument. Consider for instance a generalised version of naughts and crosses (tic-tac-toe), in which two players take turns placing naughts and crosses on some game board (not necessarily square, and not necessarily two-dimensional), with the naughts player going first, until a certain pattern of all naughts or all crosses is obtained, with the naughts player winning if the pattern is all naughts, and the crosses player winning if the pattern is all crosses. (If all positions are filled without either pattern occurring, the game is a draw.) We assume that the winning patterns for the cross player are exactly the same as the winning patterns for the naughts player (but with naughts replaced by crosses, of course).

Proposition 12 In any generalised version of naughts and crosses, there is no strategy for the second player (i.e. the crosses player) which is guaranteed to ensure victory.

Proof: Suppose for contradiction that the second player had such a winning strategy . The first player can then steal that strategy by placing a naught arbitrarily on the board, and then pretending to be the second player and using accordingly. Note that occasionally, the strategy will compel the naughts player to place a naught on the square that he or she has already occupied, but in such cases the naughts player may simply place the naught somewhere else instead. (It is not possible that the naughts player would run out of places, thus forcing a draw, because this would imply that could lead to a draw as well, a contradiction.) If we denote this stolen strategy by , then guarantees a win for the naughts player; playing the strategy for the naughts player against the strategy for the crosses player, we obtain a contradiction.

Remark 19 The key point here is that in naughts and crosses games, it is possible to play a harmless move – a move which gives up the turn of play, but does not actually decrease one’s chance of winning. In games such as chess, there does not appear to be any analogue of the harmless move, and so it is not known whether black actually has a strategy guaranteed to win or not in chess, though it is suspected that this is not the case.

Remark 20 The Hales-Jewett theorem shows that for any fixed board length, an -dimensional game of naughts and crosses is unable to end in a draw if is sufficiently large. An induction argument shows that for any two-player game that terminates in bounded time in which draws are impossible, one player must have a guaranteed winning strategy; by the above proposition, this strategy must be a win for the naughts player. Note, however, that Proposition 12 provides no information as to how to locate this winning strategy, other than that this strategy belongs to the naughts player. Nevertheless, this gives a second example in which the no-self-defeating-object argument can be used to ensure the existence of some object, rather than the non-existence of an object. (The first example was the prime number theorem, discussed earlier.)

The strategy-stealing argument can be applied to real-world economics and finance, though as with any other application of mathematics to the real world, one has to be careful as to the implicit assumptions one is making about reality and how it conforms to one’s mathematical model when doing so. For instance, one can argue that in any market or other economic system in which the net amount of money is approximately constant, it is not possible to locate a universal trading strategy which is guaranteed to make money for the user of that strategy, since if everyone applied that strategy then the net amount of money in the system would increase, a contradiction. Note however that there are many loopholes here; it may be that the strategy is difficult to copy, or relies on exploiting some other group of participants who are unaware or unable to use the strategy, and would then lose money (though in such a case, the strategy is not truly universal as it would stop working once enough people used it). Unfortunately, there can be strong psychological factors that can cause people to override the conclusions of such strategy-stealing arguments with their own rationalisations, as can be seen, for instance, in the perennial popularity of pyramid schemes, or to a lesser extent, market bubbles (though one has to be careful about applying the strategy-stealing argument in the latter case, since it is possible to have net wealth creation through external factors such as advances in technology).

Note also that the strategy-stealing argument also limits the universal predictive power of technical analysis to provide predictions other than that the prices obey a martingale, though again there are loopholes in the case of markets that are illiquid or highly volatile.

— 5. Physics —

In a similar vein, one can try to adapt the no-self-defeating-object argument from mathematics to physics, but again one has to be much more careful with various physical and metaphysical assumptions that may be implicit in one’s argument. For instance, one can argue that under the laws of special relativity, it is not possible to construct a totally immovable object. The argument would be that if one could construct an immovable object in one inertial reference frame, then by the principle of relativity it should be possible to construct an object which is immovable in another inertial reference frame which is moving with respect to the first; setting the two on a collision course, we obtain the classic contradiction between an irresistible force and an immovable object. Note however that there are several loopholes here which allow one to avoid contradiction; for instance, the two objects could simply pass through each other without interacting.

In a somewhat similar vein, using the laws of special relativity one can argue that it is not possible to systematically generate and detect tachyon particles – particles traveling faster than the speed of light – because these could be used to transmit localised information faster than the speed of light, and then (by the principle of relativity) to send localised information back into the past, from one location to a distant one. Setting up a second tachyon beam to reflect this information back to the original location, one could then send localised information back to one’s own past (rather than to the past of an observer at a distant location), allowing one to set up a classic grandfather paradox. However, as before, there are a large number of loopholes in this argument which could let one avoid contradiction; for instance, if the apparatus needed to set up the tachyon beam may be larger than the distance the beam travels (as is for instance the case in Mexican wave-type tachyon beams) then there is no causality paradox; another loophole is if the tachyon beam is not fully localised, but propagates in spacetime in a manner to interfere with the second tachyon beam. A third loophole occurs if the universe exhibits quantum behaviour (in particular, the ability to exist in entangled states) instead of non-quantum behaviour, which allows for such superluminal mechanisms as wave function collapse to occur without any threat to causality or the principle of relativity. A fourth loophole occurs if the effects of relativistic gravity (i.e. general relativity) become significant. Nevertheless, the paradoxical effect of time travel is so strong that this physical argument is a fairly convincing way to rule out many commonly imagined types of faster-than-light travel or communication (and we have a number of other arguments too that exclude more modes of faster-than-light behaviour, though this is an entire blog post topic in its own right).

As the previous discussion on displaying mathematics on the web has become quite lengthy, I am opening a fresh post to continue the topic.  I’m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion.

It’s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathML, jsMath, MathJax, OpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g.  LaTeX2HTML, LaTeXMathML, LaTeX2WP, Windows 7 Math Input, itex2MML, Ritex, Gellmu, mathTeX, WP-LaTeX, TeX4ht, blahtex, plastex, TtH, WebEQ, techexplorer, etc.).  Some of the formats are not widely supported by current software, and by current browsers in particular, but it seems that the situation will improve with the next generation of these browsers.

It seems that the tools that already exist are enough to improvise a passable way of displaying mathematics in various formats online, though there are still significant issues with accessibility, browser support, and ease of use.  Even if all these issues are resolved, though, I still feel that something is still missing.    Currently, if I want to transfer some mathematical content from one location to another (e.g. from a LaTeX file to a blog, or from a wiki to a PDF, or from email to an online document, or whatever), or to input some new piece of mathematics, I have to think about exactly what format I need for the task at hand, and what conversion tool may be needed.  In contrast, if one looks at non-mathematical content such as text, links, fonts, non-Latin alphabets, colours, tables, images, or even video, the formats here have been standardised, and one can manipulate this type of content in both online and offline formats more or less seamlessly (in principle, at least – there is still room for improvement), without the need for any particularly advanced technical expertise.  It doesn’t look like we’re anywhere near that level currently with regards to mathematical content, though presumably things will improve when a single mathematics presentation standard, such as mathML, becomes universally adopted and supported in browsers, in operating systems, and in other various pieces of auxiliary software.

Anyway, it has been a very interesting and educational discussion for me, and hopefully for others also; I look forward to any further thoughts that readers have on these topics.  (Also, feel free to recapitulate some of the points from the previous thread; the discussion has been far too multifaceted for me to attempt a coherent summary by myself.)

This week, Henry Towsner continued some model-theoretic preliminaries for the reading seminar of the Hrushovski paper, particularly regarding the behaviour of wide types, leading up to the main model-theoretic theorem (Theorem 3.4 of Hrushovski) which in turn implies the various combinatorial applications (such as Corollary 1.2 of Hrushovski). Henry’s notes can be found here.

A key theme here is the phenomenon that any pair of large sets contained inside a definable set of finite measure (such as ) must intersect if they are sufficiently “generic”; the notion of a wide type is designed, in part, to capture this notion of genericity.

— 1. Global types —

Throughout this post, we begin with a countable structure of a language , and then consider a universal elementary extension of (i.e. one that obeys the saturation and homogeneity properties as discussed in Notes 1. Later on, will contain the language of groups, and then we will rename as to emphasise this.

Recall from Notes 1 that a partial type over a set is a set of formulae (with variables for some fixed ) using as constant symbols, which is consistent and contains the theory of ; if the set of formulae is maximal (i.e. complete), then it is a type. One can also think of a type as an ultrafilter over the -definable sets; if is a type and is an -definable set given by some formula , then either lies in (in which case we write ) or lies in (in which case we write ) but not both.

When the set is small (i.e. has cardinality less than that of , which in particular would be true of consisted of union with a finite set, which is a very typical situation), then by saturation one can identify types (or partial types) with the subset of that they cut out. In particular, these sets are non-empty. Adding more formulae to a partial type corresponds to shrinking the set that they cut out, and vice versa.

However, if we have a global type – a type defined over the entire model – then one can no longer identify types with the set that they cut out, because these sets are usually empty! However, what we can do is restrict to some smaller set of constants to create a type over , defined as the set of all formulae in that only involve the constants in . It is easy to see that this is still a type, and if is small, it cuts out a non-empty set in .

A global type is said to be -invariant, or invariant for short, if the set of formulae in is invariant under any automorphism of that fixes . In particular, given any -definable set and , we see that if and only if , where is a slice of . (Indeed, this gives an equivalent definition of invariance.)

A trivial example of an invariant global type would be the type of an element (or in ). This cuts out a singleton set . This is in fact the only invariant global type that cuts out anything at all:

Lemma 1 Let be a global invariant type. Then is realisable in (i.e. is non-empty) if and only if it is realisable in (and is the type of a single element in ).

Proof: Suppose is realisable in by some . Since the formula is definable in , we see that , i.e. cuts out precisely the singleton set . As is invariant, must then be invariant under all -fixing automorphisms of . By homogeneity, this means that there is no element distinct from which is elementarily indistinguishable from over ; in other words, is the set cut out by the type of over .

By saturation, the formula together with the formulae in is not satisfiable, hence not finitely satisfiable. Thus there is a finite set of formulae in that cut out , i.e. is definable over . But as is an elementary extension of , these formulae must also be realisable in , i.e. lies in , and the claim follows.

(Because of this, one should regard the notation carefully; the set that cuts out in the model may in fact be empty, but when we write for some definable , we interpret this syntactically rather than semantically (or equivalently, that holds in all extensions of , and not just in itself.)

On the other hand, invariant global types exist in abundance:

Lemma 2 Let be a type over . Then there exists an invariant global type that refines (i.e. it contains all the formulae that does).

Proof: We view as a collection of logically consistent formulae. We enlarge this collection to a larger one by adding in the negations of all the formulae definable over that are not realisable in . Observe that this collection remains logically consistent, because any finite set of formulae in were realisable in , hence in (which is an elementary substructure). Hence, by Zorn’s lemma, one can extend to a global type .

We now claim that is invariant. Indeed, let be a sentence over that is contained in , and let be an automorphism that fixes . If is not in , then must be in (by completeness), and hence is in also, and hence must be realisable in (otherwise its negation would be in , and hence in ). But this is absurd since fixes . Thus does lie in , yielding invariance.

A major use of invariant global types for us will be that they can be used to generate sequences of indiscernibles (as defined in previous notes):

Lemma 3 Let be a global invariant type of some arity , and construct recursively a sequence by requiring for all . (This is always possible since types are satisfiable once restricted to small sets, by saturation, as discussed earlier). Then are indiscernible over , i.e. the tuples for are elementarily indistinguishable (over ) for any fixed .

Proof: This is achieved by an induction on . The case is clear since the all have type over . Now we do the case. It suffices to show that and are elementarily indistinguishable over for all .

By construction, and have the same type over , and so and are elementarily indistinguishable over . So it remains to show that and are elementarily indistinguishable.

Let be an -definable relation that contains ; we need to show that contains also.

Since and have the same type over , by homogeneity there exists an automorphism of fixing that maps to . Since realises , we see that contains the sentence , hence by invariance contains also. Since realises , we conclude , as required.

This concludes the case. The higher case is similar and is left as an exercise.

— 2. Intersections of wide types —

Now we assume that the structure is equipped with an -invariant Kiesler measure . This leads to the notion of a wide type – a type such that all the -definable sets containing this type have positive measure. Intuitively, elements of a wide type are distributed “generically” in the structure.

In the previous notes we showed that wide types can be “split” amongst indiscernables, as follows:

Lemma 4 Let be an element or tuple in , let be a wide type over for some set of constants , and let be a sequence of indiscernibles (over ) that has the same type as (over ). Then for any finite number in this sequence, one can find a type such that has the same type over as does over , for all .

We now use this lemma to show that sets defined by wide types intersect each other in a uniform fashion.

Lemma 5 Let be types over , and let , be realisations of such that and are wide. Let , be -definable sets with parameters, contained inside an -definable set of finite measure; then if and only if .

Proof: By homogeneity, there is an automorphism fixing that sends to , and maps to another element of . Thus without loss of generality we may assume .

We assume for contradiction that and .

By Lemma 2, we may extend to an invariant global type . Observe that for any , either one has for all , or one has for all (since there is a -definable set between and . Suppose first that the former option holds for some , thus there is a uniform lower bound . We now define a sequence and an indiscernible sequence as follows:

• We initialise and to be a realisation of .
• Now suppose that have been chosen with indiscernible. By Lemma 4, we can find that has the same type over that has over for all . Since , this implies that for all . (Here we use the fact that is a type-definable formula over and .)
• Now, let be a realisation of . With this construction and Lemma 3 we see by induction that is also indiscernible; now we iterate the procedure.

Let be the set , then observe from the above construction that lies in and for all . On the other hand, we claim that is uniformly bounded away from zero, this contradicts the finite measure of by the pigeonhole principle.

To see the uniform lower bound, find an automorphism fixing that maps to . By hypothesis, , thus there exists a rational such that the predicate that models is in , hence in . By invariance, the predicate is in also, hence by construction of , , and the claim follows.

Now we consider the opposite case, in which for all . Then we run the construction slightly differently: for each in turn, set to be a realisation of , then set so that . (This is possible because for any definable set containing , the -definable set contains and thus has positive measure, and so the same is true for ; now use saturation.) Then again we see that the lie in , have intersection of measure zero, and have measure uniformly bounded from below, and we again obtain a contradiction.

Now we place a group structure on , and obtain a variant of the above result:

Proposition 6 Let be types in , with wide. Suppose that , are contained in -definable sets such that has finite measure. Let be such that the type of over is wide. Assume also that the Keisler measure is translation-invariant. Then is also wide.

Proof: Suppose this is not the case, so that there exists an -definable set containing such that has zero measure. (Initially, one would need two different definable sets containing , but one can simply take their intersection.) On the other hand, as is wide, itself has positive measure. We can place in .

By using the fact that wide types over one set of constants can be refined to wide types over larger sets of constants (Lemma 2 from the previous notes), we see that we can recursively construct a sequence with wide for all . Since , we conclude from Lemma 5 that for all . On the other hand, the all have the same measure as , which is positive. Finally, the are all contained in , which has finite measure; this leads to a contradictoin.

This “generic intersection” property of translates of will be important in later arguments when creating near-groups.

The various languages and formats that make up modern web pages (HTML, XHTML, CSS, etc.) work wonderfully for most purposes, but there is one place where they are still somewhat clunky, namely in the presentation of mathematical equations and diagrams on web pages. While web formats do support very simple mathematical typesetting (such as the usage of basic symbols such as π, or superscripts such as x²), it is difficult to create more sophisticated (and non-ugly) mathematical displays, such as

without some additional layer of software (in this case, WordPress’s LaTeX renderer). These type of ad hoc fixes work, up to a point, but several difficulties still remain. For instance:

1. There is no standardisation with regard to mathematics displays. For instance, WordPress uses $latex and $ to indicate a mathematics display, Wikipedia uses <math> and </math>, the current experimental Google Wave plugins use $$ and $$, and so forth.
2. Mathematical formulae need to be compiled from a plain text language (much as with LaTeX), rather than edited directly on a visual editor. This is in contrast to other HTML elements, such as links, boldface, colors, etc.
3. One cannot easily cut and paste a portion of a web page containing maths displays into another page or file (although with WordPress’s format, things are not so bad as the raw LaTeX code will be captured as plain text). Again, this is in contrast to other HTML elements, which can be cut and pasted quite easily.
4. Currently, mathematical displays are usually rendered as static images and thus cannot be easily edited without recompiling the source code for that display. A related issue is that the images do not automatically resize when the browser scale changes; also, in some cases they do not blend well with the background colour scheme for the page.
5. It is difficult to take an extended portion of LaTeX and convert it into a web page or vice versa, although tools such as Luca Trevisan’s LaTeX to WordPress converter achieve a heroic (and very useful) level of partial success in this regard.

There are a number of extensions to the existing web languages that have been proposed to address some of these difficulties, the most well known of which is probably MathML, which is used for instance in the n-Category Café. So far, though, adoption of the MathML standard (and development of editors and other tools to take advantage of this standard) seems to not be too widespread at present.

I’d like to open a discussion, then, about what kinds of changes to the current web standards could help facilitate the easier use of mathematical displays on web pages. (I’m indirectly in contact with some people involved in these standards, so if some interesting discussions arise here, I can try to pass them on.)

A handy inequality in additive combinatorics is the Plünnecke-Ruzsa inequality:

Theorem 1 (Plünnecke-Ruzsa inequality) Let be finite non-empty subsets of an additive group , such that for all and some scalars . Then there exists a subset of such that .

The proof uses graph-theoretic techniques. Setting , we obtain a useful corollary: if has small doubling in the sense that , then we have for all , where is the sum of copies of .

In a recent paper, I adapted a number of sum set estimates to the entropy setting, in which finite sets such as in are replaced with discrete random variables taking values in , and (the logarithm of) cardinality of a set is replaced by Shannon entropy of a random variable . (Throughout this note I assume all entropies to be finite.) However, at the time, I was unable to find an entropy analogue of the Plünnecke-Ruzsa inequality, because I did not know how to adapt the graph theory argument to the entropy setting.

I recently discovered, however, that buried in a classic paper of Kaimonovich and Vershik (implicitly in Proposition 1.3, to be precise) there was the following analogue of Theorem 1:

Theorem 2 (Entropy Plünnecke-Ruzsa inequality) Let be independent random variables of finite entropy taking values in an additive group , such that for all and some scalars . Then .

In fact Theorem 2 is a bit “better” than Theorem 1 in the sense that Theorem 1 needed to refine the original set to a subset , but no such refinement is needed in Theorem 2. One corollary of Theorem 2 is that if , then for all , where are independent copies of ; this improves slightly over the analogous combinatorial inequality. Indeed, the function is concave (this can be seen by using the version of Theorem 2 (or (2) below) to show that the quantity is decreasing in ).

Theorem 2 is actually a quick consequence of the submodularity inequality

in information theory, which is valid whenever are discrete random variables such that and each determine (i.e. is a function of , and also a function of ), and and jointly determine (i.e is a function of and ). To apply this, let be independent discrete random variables taking values in . Observe that the pairs and each determine , and jointly determine . Applying (1) we conclude that

which after using the independence of simplifies to the sumset submodularity inequality

(this inequality was also recently observed by Madiman; it is the case of Theorem 2). As a corollary of this inequality, we see that if , then

and Theorem 2 follows by telescoping series.

The proof of Theorem 2 seems to be genuinely different from the graph-theoretic proof of Theorem 1. It would be interesting to see if the above argument can be somehow adapted to give a stronger version of Theorem 1. Note also that both Theorem 1 and Theorem 2 have extensions to more general combinations of than ; see this paper and this paper respectively.

It is also worth remarking that the above inequalities largely carry over to the non-abelian setting. For instance, if are iid copies of a discrete random variable in a multiplicative group , the above arguments show that the function is concave. In particular, the expression decreases monotonically to a limit, the asymptotic entropy . This quantity plays an important role in the theory of bounded harmonic functions on , as observed by Kaimonovich and Vershik:

Proposition 3 Let be a discrete group, and let be a discrete random variable in with finite entropy, whose support generates . Then there exists a non-constant bounded function which is harmonic with respect to (which means that for all ) if and only if .

Proof: (Sketch) Suppose first that , then we see from concavity that the successive differences converge to zero. From this it is not hard to see that the mutual information

goes to zero as . Informally, knowing the value of reveals very little about the value of when is large.

Now let be a bounded harmonic function, and let be large. For any and any value in the support of , we observe from harmonicity that

But from the asymptotic vanishing of mutual information and the boundedness of , one can show that the right-hand side will converge to , which by harmonicity is equal to . Thus is invariant with respect to the support of , and is thus constant since this support generates .

Conversely, if is non-zero, then the above arguments show that stays bounded away from zero as , thus reveals a non-trivial amount of information about . This turns out to be true even if is not deterministic, but is itself random, varying over some medium-sized range. From this, one can find a bounded function such that the conditional expectation varies non-trivially with . On the other hand, the bounded function is approximately harmonic (because we are varying ), and has some non-trivial fluctuation near the identity (by the preceding sentence). Taking a limit as (using Arzelá-Ascoli) we obtain a non-constant bounded harmonic function as desired.

Here is a nice version of the periodic table (produced jointly by the Association for the British Pharmaceutical Industry, British Petroleum, the Chemical Industry Education Centre, and the Royal Society for Chemistry) that focuses on the applications of each of the elements, rather than their chemical properties.  A simple idea, but remarkably effective in bringing the table to life.

It might be amusing to attempt something similar for mathematics, for instance creating a poster that takes each of the top-level categories in the AMS 2010 Mathematics Subject Classification scheme (or perhaps the arXiv math subject classification), and listing four or five applications of each, one of which would be illustrated by some simple artwork.  (Except, of course, for those subfields that are “seldom found in nature”. :-) )

A project like this, which would need expertise both in mathematics and in graphic design, and which could be decomposed into several loosely interacting subprojects, seems amenable to a polymath-type approach; it seems to me that popularisation of mathematics is as valid an application of this paradigm as research mathematics.   (Admittedly, there is a danger of “design by committee“, but a polymath project is not quite the same thing as a committee, and it would be an interesting experiment to see the relative strengths and weaknesses of this design method.)   I’d be curious to see what readers would think of such an experiment.

[Update, Oct 25: A Math Overflow thread to collect applications of each of the major branches of mathematics has now been formed here, and is already rather active.  Please feel free to contribute!]

[Via this post from the Red Ferret, which was suggested to me automatically via Google Reader's recommendation algorithm.]

Yehuda Shalom and I have just uploaded to the arXiv our paper “A finitary version of Gromov’s polynomial growth theorem“, to be submitted to Geom. Func. Anal..  The purpose of this paper is to establish a quantitative version of Gromov’s polynomial growth theorem which, among other things, is meaningful for finite groups.   Here is a statement of Gromov’s theorem:

Gromov’s theorem. Let be a group generated by a finite (symmetric) set , and suppose that one has the polynomial growth condition

(1)

for all sufficiently large and some fixed , where is the ball of radius generated by (i.e. the set of all words in of length at most , evaluated in ).  Then is virtually nilpotent, i.e. it has a finite index subgroup which is nilpotent of some finite step .

As currently stated, Gromov’s theorem is qualitative rather than quantitative; it does not specify any relationship between the input data (the growth exponent and the range of scales for which one has (1)), and the output parameters (in particular, the index of the nilpotent subgroup of , and the step of that subgroup).  However, a compactness argument (sketched in this previous blog post) shows that some such relationship must exist; indeed, if one has (1) for all for some sufficiently large , then one can ensure has index at most and step at most for some quantities , ; thus Gromov’s theorem is inherently a “local” result which only requires one to multiply the generator set a finite number of times before one sees the virtual nilpotency of the group.  However, the compactness argument does not give an explicit value to the quantities , and the nature of Gromov’s proof (using, in particular, the deep Montgomery-Zippin-Yamabe theory on Hilbert’s fifth problem) does not easily allow such an explicit value to be extracted.

Another point is that the original formulation of Gromov’s theorem required the polynomial bound (1) at all sufficiently large scales .  A later proof of this theorem by van den Dries and Wilkie relaxed this hypothesis to requiring (1) just for infinitely many scales ; the later proof by Kleiner (which I blogged about here) also has this relaxed hypothesis.

Our main result reduces the hypothesis (1) to a single large scale, and makes most of the qualitative dependencies in the theorem quantitative:

Theorem 1. If (1) holds for some for some sufficiently large absolute constant , then contains a finite index subgroup which is nilpotent of step at most .

The argument does in principle provide a bound on the index of in , but it is very poor (of Ackermann type).  If instead one is willing to relax “nilpotent” to “polycyclic“, the bounds on the index are somewhat better (of tower exponential type), though still far from ideal.

There is a related finitary analogue of Gromov’s theorem by Makarychev and Lee, which asserts that any finite group of uniformly polynomial growth has a subgroup with a large abelianisation.  The quantitative bounds in that result are quite strong, but on the other hand the hypothesis is also strong (it requires upper and lower bounds of the form (1) at all scales) and the conclusion is a bit weaker than virtual nilpotency.  The argument is based on a modification of Kleiner’s proof.

Our argument also proceeds by modifying Kleiner’s proof of Gromov’s theorem (a significant fraction of which was already quantitative), and carefully removing all of the steps which require one to take an asymptotic limit.  To ease this task, we look for the most elementary arguments available for each step of the proof (thus consciously avoiding powerful tools such as the Tits alternative).  A key technical issue is that because there is only a single scale for which one has polynomial growth, one has to work at scales significantly less than in order to have any chance of keeping control of the various groups and other objects being generated.

Below the fold, I discuss a stripped down version of Kleiner’s argument, and then how we convert it to a fully finitary argument.

– A modification of Kleiner’s argument –

Here is a variant of Kleiner’s argument, suitable for finitisation.  The starting point is to study the scalar Lipschitz harmonic functions on , defined as those functions which obey the harmonicity condition

(2)

for all , as well as the Lipschitz condition

(3)

The space of Lipschitz harmonic functions is a vector space which contains the constant functions.  If is finite, a simple application of the maximum principle shows that the constant functions are the only harmonic functions.  However, for infinite , non-constant harmonic functions exist:

Lemma 2. If is infinite and generated by , then there exists a non-constant Lipschitz harmonic function.

Proof (informal sketch only).  We give a proof which is more susceptible to finitisation than some of the more traditional proofs (e.g. the argument of Mok for the non-amenable case, and the argument of Lyons and Sullivan in the amenable case).   What we will do is find some Lipschitz almost-harmonic functions (in which (2) holds approximately rather than exactly) which are uniformly non-constant near the identity; taking a limit of such functions we will obtain the claim.

There are two cases: the “non-amenable” case in which the Laplacian has a spectral gap (i.e. a  Poincaré inequality), and the amenable case in which no spectral gap exists.  In the amenable case, one has non-trivial low-frequency functions, and one can use these to generate the desired non-constant Lipschitz harmonic function.  For instance, if with generators , one can use the low-frequency function to create the non-constant Lipschitz almost harmonic function , which in the limit yields the non-constant Lipschitz harmonic function .

In the non-amenable case, one takes a long random walk on using as the steps (and using, say, an exponential distribution for the length of the walk).   By construction, the probability distribution of this walk will have a small Laplacian; but by the spectral gap, it will have a large derivative.  Manipulating this function a bit (using convolution and normalisation) one can again obtain a non-constant Lipschitz almost harmonic function. 

The next result is the heart of Kleiner’s argument, and is based on an earlier argument by Colding and Minicozzi:

Proposition 2. If has polynomial growth, then is finite dimensional.

Proof. See my previous blog post; the main tool is a certain local Poincaré inequality exploiting the polynomial growth hypothesis which asserts, roughly speaking, that harmonic functions on a large ball do not fluctuate much on smaller balls.  The argument gives an explicit upper bound on the dimension of in terms of the growth order .  When one is assuming only polynomial growth at a single scale , the argument yields that any Lipschitz harmonic (or almost harmonic) functions will have a “determinant” which is extremely small, where the determinant is taken with respect to a suitable quadratic form on (defined using a scale slightly smaller than ).

The group acts on by left translations, in a manner which preserves the Lipschitz norm, which becomes a genuine norm on the finite-dimensional space after quotienting out the constants.  From Lemma 1 and Proposition 2, one thus obtains a non-trivial finite-dimensional isometric representation of .  (The precise formulation of “non-trivial” is “infinite image”, which comes from the basic fact that non-constant harmonic functions must have infinite image, thanks to the maximum principle.)  Since has polynomial growth, the image of in the isometry group also has polynomial growth.

At this point one could apply the Tits alternative and conclude that the image of is virtually solvable, but we wish to avoid the use of this alternative.  Instead, we use the basic observation (also used, for instance, in the proof of the Solovay-Kitaev theorem) that if two linear transformations are very close to the identity, then their commutator is even closer still to the identity.  If we take a few transformations in the image of that are close to the identity and take a few commutators, we either end up at the identity, or we end up with a lot of transformations at widely differing distances from the identity.  The latter case contradicts the polynomial growth hypothesis, so we can conclude that the transformations near the identity form a solvable group.  Since the isometry group of a finite-dimensional space is compact, this implies that the image is virtually solvable.  Since the image is also infinite, we conclude that the image has a finite index subgroup with an infinite abelianisation, and thus has a finite index subgroup with an infinite abelianisation.

An elementary splitting argument of Gromov then tells us that the kernel of that abelianisation has one order less growth (at least) than itself.  Applying Gromov’s theorem as an induction hypothesis, this kernel is itself virtually nilpotent.  After passing to a finite index subgroup to get rid of the “virtually” adjective, we conclude that G contains a finite index subgroup whose kernel of the abelianisation is nilpotent; in particular, G is virtually solvable.

To conclude, one uses an elementary result of Milnor and Wolf that establishes Gromov’s theorem in the solvable case.  (Ultimately, things boil down to showing that the only linear transformations on the lattice which exhibit polynomial growth are virtually unipotent; this in turn boils down to a classical result of Kronecker that the only algebraic integers whose Galois conjugates all lie on the unit circle are the roots of unity.)  This completes the proof sketch of Gromov’s theorem.

It turns out that all of the above steps can be finitised.  Instead of working with Lipschitz harmonic functions on infinite groups, one works instead with Lipschitz almost harmonic functions on very large groups.  One is then not working with a finite dimensional vector space , but instead with a class of functions which is closely approximated by a finite-dimensional space.  Applying a Gram-Schmidt type procedure, we now get an approximate finite-dimensional representation of the group (in which the homomorphism property contains small errors), but the elementary Solovay-Kitaev type argument can handle this sort of noise, and one soon finds that the image of in this representation behaves as if it is virtually solvable; unpacking what this means, it implies that approximately Lipschitz harmonic functions on are approximately constant in some iterated commutator of a subgroup of .  (It is instructive to work this type of phenomenon out by hand for, say, the Heisenberg group.)  One can then quotient out this direction to reduce the order of growth.  The only remaining task is to finitise the Milnor-Wolf theory, but this is relatively straightforward (for instance, instead of Kronecker’s theorem, one uses lower bounds on Mahler measure, such as the bound due to Dobrowolski).

At UCLA we just concluded our third seminar in our reading of “Stable group theory and approximate subgroups” by Ehud Hrushovski. In this seminar, Isaac Goldbring made some more general remarks about universal saturated models (extending the discussion from the previous seminar), and then Henry Towsner gave some preliminaries on Kiesler measures, in preparation for connecting the main model-theoretic theorem (Theorem 3.4 of Hrushovski) to one of the combinatorial applications (Corollary 1.2 of Hrushovski).

As with the previous post, commentary on any topic related to Hrushovski’s paper is welcome, even if it is not directly related to what is under discussion by the UCLA group. Also, we have a number of questions below which perhaps some of the readers here may be able to help answer.

Note: the notes here are quite rough; corrections are very welcome. Henry’s notes on his part of the seminar can be found here.

(Thanks to Issac Goldbring for comments.)

— 1. More remarks on universal models —

Recall from the previous seminar that if we have any structure of a language (e.g. could be a group, and the language of groups), then we can (assuming some set-theoretic axioms, such as the existence of inaccessible cardinals) obtain an elementary extension of , enjoying the following properties:

• (i) (Saturation) If has strictly smaller cardinality than , then every partial type over is realisable in ; and
• (ii) (Homogeneity) If is as above and are elementarily indistinguishable over , then there is an automorphism of that maps to .

Another way of phrasing saturation: if one has a “small” family of formulae (where “small” means “having cardinality less than that of “) which is finitely realisable in (i.e. for any finite collection of these formulae, one can find obeying these formulae), then it is realisable (one can find obeying all of the formulae). Equivaqlently the topology on generated by the definable sets (which form a clopen base) is compact.

Remark 1 has the following “universal” property: any small model of the theory of (i.e. any structure elementarily equivalent to ) can be embedded into . Indeed, if one throws in all the elements of as constants, the model can be described as a partial type over the small set , and one simply applies saturation to obtain the embedding.

(A side remark: in Hrushovski’s paper, it is not completely clear whether one is working with the universal model (in which saturation holds over all small sets ) or merely a countably saturated one. If it is the latter, then the above remark does not quite hold, but it seems that one can get around this by replacing “small model” with “small elementary substructure of ” throughout the paper. But this seems to not make a significant difference to the substance of the paper.)

Usually we fix a universal structure , and abbreviate as . (In the combinatorial applications, will be a group, and we will thus call it .)

Remark 2 Universal models have the following useful cardinality gap property: if is a definable set, then either is finite, or (i.e. is large); there are no definable sets of intermediate cardinality. Proof: let be a defining formula for . If is infinite, then the formula together with the formulae for every are finitely realisable, hence realisable by saturation, but this is absurd. Note that this argument in fact shows that any -definable set (i.e. a partial type) is either finite or large. For -definable sets over some infinite set of constants , it was claimed that such sets either have cardinality less than or equal to , or are large, but I didn’t check this.

(Question: if one only assumes countable saturation (which may be the case in Hrushovski’s paper), then one has a smaller cardinality gap: definable sets can be finite or uncountable. It wasn’t clear to us whether the rest of the paper would work if one just had this gap.)

Now we obtain an analogous cardinality gap for equivalence relations.

Lemma 1 Suppose the language is at most countable. Let be a -definable equivalence relation on (one could also consider equivalence relations on ). Then either the cardinality of the classes is at most (the cardinality of the continuum), or is .

Proof: Let be one of the defining relations of the equivalence relation , which we can assume to be symmetric. By the greedy algorithm, we see that one of the following two statements must be true:

• There exists finitely many such that for every , is true for at least one .
• There exists a countable sequence such that fails for all .

Suppose the latter possibility holds. Then by transfinite induction (using the least ordinal with cardinality ) and saturation, we can obtain a family of cardinality such that fails for all . This implies that has equivalence classes, and we are done. Thus the only case remaining is if every defining relation enjoys the first property. But then we see that the equivalence class that an element belongs to is determined by the precise set of obeying for each , and this implies that the number of classes is at most .

We remark that is best possible. Indeed, consider the integers in the language of groups, and take a universal extension of these integers. Consider the relation defined by cosets of the group (i.e. if and agree modulo for every (standard) natural number ). Then this is a -definable relation, but the set of equivalence classes is equivalent to the profinite integers , which have the cardinality of the continuum.

Equivalence relations whose set of classes have cardinality at most are known as bounded equivalence relations. (It may be that one does not actually need the above dichotomy to establish the results in Hrushovski’s paper, and just take the above statement as the definition of a bounded equivalence relation.)

Observe that the proof of the above lemma shows that if a bounded equivalence relation is definable rather than merely -definable, then the number of classes is finite. Thus bounded can be viewed as a generalisation of “finite”.

The boundedness of a certain equivalence relation (created by a certain stabiliser group) is essential in ensuring that we can define a certain locally compact “logic topology” on the quotient space, which is apparently what allows the Gleason-Yamabe theory of topological groups to kick in; presumably we will see more of this in later seminars. (This idea also appears in previous work, such as this paper by Hrushovski, Peterzil, and Pillay.)

— 2. Keisler measures —

To obtain combinatorial results, Hrushovski applies the contradiction and compactness method: assume that a combinatorial result fails, obtain a sequence of counterexamples to that result, and extract some sort of limiting object to which an infinitary theory (in this case, stability theory) can be applied. In this section we discuss what happens to the combinatorial concept of (normalised) cardinality along such a limit; what one gets at the end is an object known as a Kiesler measure.

Suppose we have a finite set in a group . Then given any other set in , we can define a normalised counting measure of by the formula (this measure could be infinite if is infinite). More generally, given , one can define . We will often omit the superscript and use for all the measures on simultaneously. Thus for instance one has , and if and .

Now suppose one has a sequence of groups, each with a finite set . Then we have a sequence of normalised counting measures on (and on powers ). We can then take an ultralimit of all of these objects, obtaining a group (the ultrapower of the ), a subset of (the ultrapower of the ), and a “measure” , which assigns to every subset of a non-standard real number (or ) . Thus for instance . In practice we will be working inside (or small powers of , such as ), which by hypothesis will have bounded measure, so the case will not be relevant.

In applications, the will have bounded doubling, thus is bounded in ; taking ultraproducts, we see that is bounded also (i.e. less than a standard real number).

Now we want to extend this ultrapower to a universal model . To do this, we have to specify the language . will of course contain the language of groups, but if that is all that the language contains, then one cannot refer to or in this language. To add in is easy: one simply adds a unary predicate for membership in , so that “” is now a formula in . To add is a bit trickier: to keep the language countable, one is not allowed to measure all sets in the model, and also one does not want to work with non-standard real numbers. So instead, what one does is add predicates that are interpreted as “” for any definable set with parameters , and any (standard) rational number . (Note that by adding these predicates, one enlarges the space of possible definable sets; but one can iterate this countably many times to deal with this apparent circularity.) We thus obtain a countable set of predicates (even after iteration). With all these predicates, the language is now capable of interpreting as a standard (extended) real number (the supremum of all rational for which ); when the model is the ultrapower , this standard real number is simply the standard part of the non-standard real number .

[A slight subtlety: when one collapses to the standard real numbers, the predicate now deviates very slightly from its original interpretation "". What is now true is that this predicate is implied by , and in turn implies "", just as the assertion that a non-standard number is larger than is implied by the standard part being larger than , and in turn implies .]

With this language , we can now pass to a universal model . This is a group with a set , and a real number assigned to any set in or definable with parameters. It inherits all the elementary properties of , for instance is finitely additive and one has and . (One can use compactness and the Carathéodory extension theorem to extend to a -additive measure on the “Borel -algebra” generated by the definable sets; this is basically the Loeb measure construction, but we will not need it here.)

By construction, is a Keisler measure – a finitely additive measure on the definable sets. By construction, it is also -invariant – i.e. it is invariant under all automorphisms of that fix . One consequence of this is that the quantity depends only on the type of over . Actually, there is a stronger statement – the quantity depends continuously on using the topology generated by the -definable sets (by using the predicates ). Because of this, we call an -definable Keisler measure.

Remark 3 One does not need to pass through the non-standard real numbers here; instead, one can interpret directly as in the finite models , take an ultralimit (or some other theory that contains all the statements that are true in all but finitely many of the ) and then take a universal extension, and then reconstruct as the supremum of all for which holds. I find the temporary non-standard interpretation of to be helpful though in understanding why ends up deviating slightly from .

— 3. Wide types —

Recall that a partial type in over some set of constants is a set of -ary formulae using , which then defines a -definable (over ) subset of , being the set of all -tuples that realise those formulae. A complete type is a maximal such set of formulae (or a minimal -definable set, equivalently), or equivalently an orbit of the automorphism group of .

Note that given a complete type and a definable set, the type either falls entirely inside the set or is disjoint from it. (Indeed, a complete type can be thought of as an ultrafilter on the algebra of definable sets.) In practice, all the types we will consider will lie inside a definable set such as , which will have finite measure in applications.

Now for a key concept. A (partial or complete) type is wide if the set it defines it cannot be contained in a definable set of zero measure. More generally, a type is wide over a set of constants if it cannot be contained in an -definable set of zero measure. Wide sets are thus somewhat “generic” or “Zariski-dense” in some sense.

Lemma 2 Every wide partial type over can be refined to a wide complete type over .

Proof: Let be a wide partial type over . By finite additivity of , cannot be covered by finitely many -definable sets of measure zero, thus by saturation (compactness), cannot be covered by all of such sets at once. If we set to be with all the -definable sets of measure zero, is then also a partial type which is wide (since it is disjoint from, and hence not contained in, any -definable set of measure zero). If we then complete to any complete type (e.g. by taking an element of and computing its type), this type is then also wide.

A typical application of this lemma will be to start with a wide complete type over some set of constants, and then enlarge the set of constants, thus turning the complete type back into a partial type; but if one can show that the partial type remains wide, then one can refine it back to a wide complete type.

— 4. Types and group operations —

Types are well behaved with respect to group operations. For instance, if is a partial type over , and , then the shift of (defined by taking the set in or defined by and shifting it by ) is a partial type over ; if contained and was a complete type, then so is .

In a similar vein, if is a partial or complete type over , then is a partial or complete type over .

Slightly less obvious is that if are partial types over , then the product set is also a partial type over . Indeed, for any -definable sets containing , we see that is contained in the set , which is -definable (being cut out by the formula ““). On the other hand, if lies in all the , i.e. for each one can find realisations of the formula ““, then (as the intersection of definable sets is still definable) this whole family of formulae is finitely realisable, and thus (by saturation) realisable, which means that . Thus we have exhibited as a finite type.

— 5. Indiscernibles —

We already have a notion of elementary indistinguishability: two -tuples are elementarily indistinguishable (over some set of constants ) if every formula involving obeyed by is also obeyed by and vice versa (i.e. have the same type, or equivalently (by homogeneity) there is an automorphism that maps to ).

We can build upon this notion: a sequence is indiscernible (over ), if for every , the -tuples for are elementarily indistinguishable; thus the are indistinguishable, but also every ordered pair for are indistinguishable, and so forth. In practice, indiscernibles will be constructed by lots of Ramsey theory, and are sort of a way of pre-emptively doing all the Ramsey theory at once.

Lemma 3 Let be an -definable set for some , and for each , let be the slice (note that this is a -definable set). We assume that all the are contained in an -definable set of finite measure. Let be such that , and let be an indiscernible sequence (over ) with the type of (i.e. they are all elementarily indistinguishable from . Then any finite number of the have non-empty intersection (in fact their intersection has positive measure).

Proof: We establish this by induction. Suppose that we already know that

for all (note that this is already true for by hypothesis and indiscernability). We claim that

for also. Note by indiscernability that these measures do not actually depend on the choices of . Suppose for contradiction that we had

for one, and hence (by indiscernability) for all . In particular, this shows that the sets

are disjoint modulo null sets for . On the other hand, these sets have uniformly positive measure by induction hypothesis and indiscernability. Thus their union has unbounded measure, which contradicts the hypothesis that they all lie in a set of finite measure.

A remark: the above argument was phrased using the Kiesler measure , but the same argument also works for a more general structure known as a -ideal. We’re not clear yet whether this additional generality will be necessary for the applications at hand.

Now we combine wide types and indiscernables, showing that wide types can be perturbed to “look the same” with respect to any finite number of indiscernibles:

Lemma 4 Let be an element or tuple in , let be a wide type over for some set of constants , and let be a sequence of indiscernibles (over ) that has the same type as (over ). Then for any finite number in this sequence, one can find a type such that has the same type over as does over , for all .

Proof: To abbreviate notation we suppress “over ” in everything that follows. (We’ll probably use this convention again in later seminars.)

Let be a formula obeyed by and , then the definable set cut out by is such that the slice has positive measure (since is wide). By the previous lemma, is thus non-empty, so we can find an such that is true for all . At present depends on ; but by using saturation one can make independent of , and the claim follows.

The polymath1 project has just uploaded to the arXiv the paper “A new proof of the density Hales-Jewett theorem“, to be submitted shortly.  Special thanks here go to Ryan O’Donnell for performing the lion’s share of the writing up of the results, and to Tim Gowers for running a highly successful online mathematical experiment.

I’ll state the main result in the first non-trivial case for simplicity, though the methods extend surprisingly easily to higher (but with significantly worse bounds).  Let be the size of the largest subset of the cube that does not contain any combinatorial line.    The density Hales-Jewett theorem of Furstenberg and Katznelson shows that .  In the course of the Polymath1 project, the explicit values

were established, as well as the asymptotic lower bound

(actually we have a slightly more precise bound than this).  The main result of this paper is then

Theorem. ( version) .

Here is the inverse tower exponential function; it is the number of times one has to take (natural) logarithms until one drops below 1.  So it does go to infinity, but extremely slowly.  Nevertheless, this is the first explicitly quantitative version of the density Hales-Jewett theorem.

The argument is based on the density increment argument as pioneered by Roth, and also used in later papers of Ajtai-Szemerédi and Shkredov on the corners problem, which was also influential in our current work (though, perhaps paradoxically, the generality of our setting makes our argument simpler than the above arguments, in particular allowing one to avoid use of the Fourier transform, regularity lemma, or Szemerédi’s theorem).   I discuss the argument in the first part of this previous blog post.

I’ll end this post with an open problem.  In our paper, we cite the work of P. L. Varnavides, who was the first to observe the elementary averaging argument that showed that Roth’s theorem (which showed that dense sets of integers contained at least one progression of length three) could be amplified (to show that there was in some sense a “dense” set of arithmetic progressions of length three).  However, despite much effort, we were not able to expand “P.” into the first name.  As one final task of the Polymath1 project, perhaps some readers with skills in detective work could lend a hand in finding out what Varnavides’ first name was? Update, Oct 22: Mystery now solved; see comments.