Finite and Infinite 有涯与无尽math.sjtu.edu.cn/faculty/ykwu/data/Talk/20190627.pdf ·...

起: 呈示承: 巩固转: 发展合: 结束

Finite and Infinite有涯与无尽

Yaokun Wu (吴耀琨)Shanghai Jiao Tong University

June 27, 2019

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Finite and infinite

Infinite/infinitesimal in space/time draws attentions of many poets:无边落木萧萧下，不尽长江滚滚来 – 杜甫，登高

Finiteness appears in poems as well, though not so much:世路虽多梗，吾生亦有涯 – 杜甫，春归

How do we play with finite and infinite in mathematics?

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Finite and infinite

Infinite/infinitesimal in space/time draws attentions of many poets:无边落木萧萧下，不尽长江滚滚来 – 杜甫，登高Finiteness appears in poems as well, though not so much:世路虽多梗，吾生亦有涯 – 杜甫，春归

How do we play with finite and infinite in mathematics?

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Finite, or infinite?

The starry sky suggests to us an image of infinity; but in our dailylife we only come across finite objects.

God created infinity, and man, unable to understand infin-ity, had to invent finite sets. – Gian-Carlo Rota

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https://www.vangoghgallery.com/catalog/Painting/508/Starry%20Night.html

http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/rota-gian-carlo.pdf


How round is a circle? (割圆术)

Human eyes/brains recognize the perfect image of a circle from thenature:

大漠孤烟直，长河落日圆。–王维

Archimedes (287-212 BC) used polygons as finite approximationsof the circle in order to compute the area of a disk.

• Disk: Infinitely many points at a fixed distance to the originand their convex hull;

• Polygon: Finitely many points at a fixed distance to the originand their convex hull.

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Turning stone to bread

People cannot do or make or plan infinity. A human hastwo hands: one hand is in infinity, the other hand is inthe finite real world. I think that the real task of themathematician is to somehow connect these two.Consciously or unconsciously, what mathematicians do isa finitization of infinity. – Heisuke Hironaka (広中平祐)

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http://www.ams.org/notices/200509/fea-hironaka.pdf

https://www.heidelberg-laureate-forum.org/blog/laureate/heisuke-hironaka/


Continuous, or discrete?

Continuous analysis and geometry are just degenerate ap-proximations to the discrete world, made necessary bythe very limited resources of the human intellect. –Doron Zeilberger

Finite objects are often imperfect approxima-tions of much nicer infinite objects. The con-nection between finite and infinite structures isuseful in both directions. – Balázs Szegedy,From graph limits to higher order Fourier analysis, 2018.

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http://sites.math.rutgers.edu/~zeilberg/Opinion57.html

https://eta.impa.br/dl/047.pdf


Intersecting family: From bounded to unbounded

Helly property: Let C be a fmaily of convex sets in Rd. If everyd + 1 elements of C has nonempty intersection, then elements of Cpossess a common point.

A quiz: Let D be a family of sets in which there is a set of size atmost d. If every d + 1 members of D intersect, then so does D.

Compactness property: Let C be a family of closed subsets in acompact topological space. If every finite subset of C hasnonempty intersection, then so does C.

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Dirichlet box principle for partition regularity

When you see 5 pigeons flying into 2 holes, you know that thereexists one hole which has at least 3 pigeons in it.

If you partition infinitely many objects into a finite number ofparts, you expect that one part should be still large enough to havean interesting structure.

• B.L. van der Waerden 1927: For each finite coloring of N,there are arbitrarily long monochromatic arithmeticprogressions.

• Neil Hindman 1974, Vitaly Bergelson 2003: For each finitecoloring of N, there are sequences of increasing positiveintegers a = (an)n∈N and b = (bn)n∈N such that the setFP (a) ∪ FS(b) is monochromatic.

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http://www-history.mcs.st-andrews.ac.uk/Biographies/Van_der_Waerden.html

http://nhindman.us/

https://people.math.osu.edu/bergelson.1/


Integral/Measure: Beyond ‘good’ data set

In the perfect world, you are happy with doing integrations like∫ ∞0 exp

(−x2

)dx or

∫ π0 sin x dx.

In the real world, you need to aggregate the opinions of persons onthe shape of a tree, ranking of some candidates, or average a set ofmetric spaces of not necessarily the same size.

You should already know the limit of a sequence of numbers. Youmay like to know that mathematicians have defined ultralimit totalk aboout the limit of a sequence of metric spaces, generalizingthe notion of Gromov-Hausdorff convergence of metric spaces.

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In order to apply results and ideas from continuous math-ematics to discrete settings, there are basically two ap-proaches. One is to directly discretise the arguments usedin continuous mathematics,... The other is to constructcontinuous objects as limits of sequences of discrete ob-jects of interest...Roughly speaking, one can divide them into three cate-gories: Topological and metric limits. Categorical limits.Logical limits. ... Ultraproducts are not the only logi-cal limit in the model theorist’s toolbox, but they are oneof the simplest to set up and use, and already suffice formany of the applications of logical limits outside of modeltheory. – Terence Tao, Ultraproducts as a bridge betweendiscrete and continuous analysis, 2013.

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https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/


Finite and infinite: Quantity/hard and quality/soft

There is a close relationship between finitary (or ‘hard’, or‘quantitative’) analysis, and infinitary (or ‘soft’, or ‘qual-itative’) analysis. One way to connect the two types ofanalysis is via compactness arguments; such argumentscan convert qualitative properties (such as continuity) toquantitative properties (such as bounded), basically be-cause of the fundamental fact that continuous functionson a compact space are bounded. – Terence Tao, HigherOrder Fourier Analysis, AMS, 2012.

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From Marc to me

• Marc Troyanov has discussed combinatorics of convex bodies1 (Integral Geometry, Geometric Probability) in last 30minutes. Let me address combinatorics of bigness in this slot.

• The main object of the talk of Marc is the good numericalvaluations which measure the sizes of polyconvex subsets. Myhero will be valuations which indicate qualitatively thelargeness of subsets of any set.

1According to Rota, “convexity is continuous combinatorics”.12 / 47


How to measure bigness?

Ana Pires, Hospitality at the Hilbert Hotel: How big is infinity?2016.

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https://www.ias.edu/ideas/2016/pires-hilbert-hotel


Ultrafilter (超滤子)

Ultrafilter provides a way of constructing the limit and measuringthe bigness. It is a good bridge between infinite/continuous andfinite/discrete.

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Origins of filter and ultrafilter

The theory of filters was developed by Henri Cartan in 1937.Nicolas Bourbaki further developed the theory in amazing details intheir textbook on general topology in 1949. The notion ofultrafilter was suggested by Hungarian mathematician FredericRiesz in his ICM talk at 1908 and rediscovered and studied a lot byPolish mathematician Alfred Tarski and his school in 1930.

吴文俊大事记

1919 年生于上海市1936 年由正始中学毕业，获得奖学金，指定报考交通大学数学系1949 年夏去巴黎，跟随 H. Cartan 继续拓扑学研究

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http://www.amss.cas.cn/wwj/xsly/201705/P020170507440546478833.pdf

http://www.math.sjtu.edu.cn/


Filter on Boolean lattice

A filter on X is a subset of 2X \ {∅} which is closed under takingsuperset and finite intersection. For a topological space X and apoint x ∈ X, the set of all neighborhood of x is a filter.An ultrafilter is a maximal filter; namely an ultrafilter on a set X isa subset of 2X which is maximal with respect to the finiteintersection property. Given an ultrafilter U ∈ 2X , we will think ofA ∈ U as a big set and B < U as a small set.A principal ultrafilter is an ultrafilter which consists of all setscontaining a common element. An ultrafilter is free if it is notprincipal.

• There is a free ultrafilter on X if and only if X is an infiniteset (assuming Axiom of Choice).

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Pigeonhole property

The bigness defined by ultrafilter is robust under partitioning:

Pigeonhole property: If U is an ultrafilter on X and X ispartitioned into a finite number of parts, then exactly one part isBIG.

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Ultrafilter as a voting system

Take a positive integer n ≥ 3. Suppose that in an election thereare n candidates c1, . . . , cn and a set X of voters. Each votermakes a ranking of the candidates (a permutation, or a directedpath on n labelled vertices). How to aggregate the opinions of Xso that the following hold:

• if all the voter enter the same ranking, the outcome should bethat one;

• whether a candidate a precedes candidate b depends only ontheir order on the different ranking lists of the individualvoters.

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Arrow’s Impossibility Theorem

Each such voting system must correspond to an ultrafilter U on Xsuch that the outcome is a permutation π if and only if the set ofthose voters whose ranking is π is a big set, namely a set in U .

• If X is finite, since there only exist principal ultrafilters on it,there must be a dictator among X in any scientific votingsystem!

• If X is infinite, by the pigeonhole property, every freeultrafilter does give a voting system without dictators!

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Ramsey theory and funny ultrafilters

Let G be a nonempty subsets of 2N. We think of a subset of Nwhich ocntains any element of G a large set. We call G partitionregular provided for any partition of N into a finite number of partswe can find a large part.

A simple but important observation by Hindman: The hypergraphG is partition regular if and only if there exists an ultrafilter U on Nsuch that every member of U contains a hyperedge G ∈ G.

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In the context of Shannon entropy, partition, information andcoimages of functions are dual notions of subsets, size and imagesof functions 2.

Can the observation of Hindman be explained in this line ofthoughts?

2J.P.S. Kung, G.-C. Rota, C.H. Yan, Combinatorics: The Rota Way,Cambridge University Press, 2009, p. 40.

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Model theory and ultraproduct

The ultraproduct is formed by taking the cartesian product of atleast a countable number of sets and identifying those elementsthat agree on a “large”set of sets in the cartesian product.

Modern model theory has Alfred Tarski (1901-1983) and AbrahamRobinson (1918-1974) as main founders. Ultraproductconstruction provides a natural proof of the following majortheorem of model theory:Compactness Theorem: A set of sentences Σ of a language has amodel if and only if every finite subset of Σ has a model.

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http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Halper.pdf


Elementary calculus VS 高等数学

Nonstandard analysis grew out of the attempt ofAbraham Robinson to resolve the contradictions posed byinfinitesimals within calculus.

Ultraproduct construction defines equivalence classes in RN whichare called hyperreals. The process of extending reals to hyperrealsin nonstandard analysis enables some mathematicians call calculuselementary calculus.

The relationship between ultralimit analysis and nonstandardanalysis is analogous to that between measure theory andprobability theory.

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http://www-history.mcs.st-and.ac.uk/Biographies/Robinson.html

https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf

http://susanka.org/Notes/Hyperreals.pdf

https://www.math.wisc.edu/~keisler/calc.html


复杂事物其实是简单事物的叠加投影。– 広中平祐, 创造之门

• The set of all ultrafilters on a semigroup has nice algebraicand topological structures which help us understand manydynamical and combinatorical properties.

• Filter and ultrafilter can be defined for general posets.

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Translation-invariant valuation on Z

Can we say with confidence that an integer has probability 12 for

being even?

Can we define a valuation µ on subsets of Z such that thefollowing hold?

• µ(A) ∈ [0, 1] for all A ⊆ Z;• µ(Z) = 1;• valuation: µ(A) + µ(B) = µ(A ∪ B) + µ(A ∩ B);• shift invariance: µ(A) = µ(A + n) for all n ∈ Z and A ⊆ Z;• µ(even integers)= 1

2 .

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https://arxiv.org/abs/1806.00053


Banach limit: Limit along an ultrafilter

Let U be an ultrafilter on N. For a bounded sequence of realnumbers (xn)n∈N, its limit along U 3 is the unique real number xsuch that {n | d(x, xn) < ϵ} ∈ U for all ϵ > 0.

Let yn = |A∩[−n,n]|2n and let µ(A) be the limit of (yn)n∈N along U .

If U is principal, say U = U5, then µ(A) = y5 and so we are notreally seeing the structural limit of A. However, if A is free on N,we do find that µ is a valuation satisfying all the requirementslisted in the previous slides.

3In general, for any ultrafilter U on a set X and any map f from X to acompact Hausdorff space Z, we can talk about the unique limit point of f in Zalong U .

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Measure and integral

If probability indicates your degree of belief, an ultrafilter is aprobability measure for fundamentalists.

• Each U ⊆ 2X naturally corresponds to its indicator function1U : 1U (A) = 1 if A ∈ U and 1U (A) = 0 if A ∈ 2X \ U .

• U is an ultrafilter if and only of µ = 1U is a {0, 1}-valuedvaluation (finitely additive measure).

• If U is a principal ultrafilter, then µ is a point measure.• If U is a free ultrafilter on a countable set X, then µ is

additive but not σ-additive.• The limit of a function on X along the ultrafilter U is nothing

but the integral of the function f in the measure space (X, µ).

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Products of compact spaces

Andrey Nikolayevich Tychonoff 1930: The product of any family ofcompact topological spaces is still compact.Proof.Step 1: By the pigeonhole property, along every ultrafilter on theproduct space X, the identity map has a limit. Step 2: LetU = {Ui} be an open cover of X. If U has no finite cover, thenthe set of complements of those finite unions of elements of U iscontained in an ultrafilter. Applying the conclusion in Step 1 tothis ultrafilter to yield a contradiction. □

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History of Tychonoff’s Theorem

Although many proofs in point-set topology sort of workby themselves, guided by intuition and oiled by a cun-ning terminology and spatial intuition – the proof of theTychonoff theorem is not one of these. – Klaus Jänich,Topology, Springer, 1984.

A.N. Tychonoff claimed his theorem in 1935 without proof,mentioning that a proof can be made as the one he gave for aproduct of closed intervals in 1930. An explicit proof was given byEduard Čech in 1937. The proof using ultrafilters was found byHenri Cartan in 1937.

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https://ncatlab.org/nlab/show/Tychonoff+theorem#Tychonoff35


AoC, accept or not accept?

The existence of ultrafilter is guaranteed by the Axiom of Choice.Tychonoff Theorem is equivalent to the Axiom of Choice.

AoC: : For every surjection f : A → B of sets, there is a functionσ : B → A (a section), such that f ◦ σ : B → B is the identitymap.

AoC for finite set B is trivial. But this axiom for infinite set B isreally a miracle.

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What is your choice?

(1) Is it possible that 1 + 1 = 1?(2) Is there a set A ⊆ R2 which intersects every straight line in

exactly two points?(3) Assume that every finite subgraph of a graph has chromatic

number 4. Can the graph itself be 4-colored?(4) Does every graph admit a bipartition of its vertex set in which

each vertex has at least as many neighbors in the other classas in its own?

(5) If every finite subset of a matroid is representable, is thematroid itself representable?

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A quick answer via Boolean algebra

Surely, in an idempotent semiring, say tropical semiring((max, +)-semiring) or even Boolean semiring, we have 1 + 1 = 1.

If you think of Boolean semiring unnatural, please note that ClaudShannon changed our world in his master thesis, which may be themost famous master’s thesis in last century, by using thisparadoxical Boolean semiring as the basis of digital circuit designtheory.

George Boole called the axiom of Boolean algebras the ‘laws ofthoughts’. Birkhoff pointed out that each finite Boolean algebra isisomorphic to the power set of a finite set. This led John vonNeumann call Boolean algebra ‘pointless set theory’. GeneralBoolean algebras may not be isomorphic to 2X for some set X butcan be represented by clopen subsets of Stone spaces.

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http://www.historyofinformation.com/detail.php?entryid=745

http://math.uchicago.edu/~may/REU2015/REUPapers/deOliveira.pdf

https://terrytao.wordpress.com/2009/01/12/245b-notes-1-the-stone-and-loomis-sikorski-representation-theorems-optional/


Ultrafilter and Boolean algebra

The smallest Boolean algebra, denoted by 2, consists of twoelements 0 and 1. You can think of it as the power set of asingleton set.

The set of ultrafilters on X, U(X), is nothing but the set off−1(1), where f ranges through all Boolean algebrahomomorphisms from 2X to 2.

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Paradoxical partition

A structure is paradoxical if it can be partitioned into several parts,each of them being equivalent to the original structure in somesense.

Vitali set 4 is the earliest construction of unmeasurable sets, surelyalso under the assumption of AoC. Based on Vitali sets, FelixHausdorff showed that one can chop up a unit interval intocountably many pieces and fit them together to make an intervalof length two. This demonstrates, maybe more convincingly, that1 = 1 + 1!

4This is named after the Italian mathematician Giuseppe Vitali (Do notmispell it as Vitaly). He is famous for the Vitali Covering Lemma.

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http://math.sjtu.edu.cn/conference/Bannai/2017/talk.php?20170912B

http://www-history.mcs.st-andrews.ac.uk/Biographies/Vitali.html

http://logicandanalysis.org/index.php/jla/article/viewFile/143/54


Banach-Tarski Paradox (BTP)

BTP (1924): It is possible to decompose a ball in R3 into six 5

pieces which can be reassembled by rigid motions to form two ballsof the same size as the original. The proof relies on the fact thatSO(3) contains the free group of rank 2 as a subgroup.

My goal here today is to state the BTP carefully, give youan idea of how it is proven, and (along the way) explainwhy this geometric statement is of interest to analysts andwhy an algebraist is talking about it. It turns out that thestatement is essentially equivalent to the Axiom of Choice,but that it has interesting analytical consequences, andthe main idea of the proof is algebraic. – J. Leuschke,The Banach-Tarski paradox.

5The minimum number of such pieces is 5, as determined by Robinson.35 / 47

http://www.leuschke.org/uploads/Research/Banach-Tarski.pdf


Is it true that no compact metric space is paradoxical (w.r.t.isometries) using Borel pieces?

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Unfriendly partition

• Every finite graph admits an unfriendly partition of its vertexset.

• An uncountable graph needs not have an unfriendly partitionof its vertex set.

ConjectureEvery countable graph admits an unfriendly partition of its vertexset.

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http://www.openproblemgarden.org/op/unfriendly_partitions


Topological structure

The Stone-Čech compactification βX of a (locally compactHausdorff) topological space X is a compact topological spacecontaining X as a subspace such that

βX∃!βf

!!C

CC

C

Xf

//?�

id

OO

K

holds for any continuous map from X to any compact space K.

If we associate a set X with the discrete topology, what is βX?

It coincides with U(X), the set of all ultrafilters on X, in whichevery x ∈ X corresponds to the principal ultrafilter Ux. A setV ⊆ βX is open if and only if it is a union of some sets of the formA for A ⊆ X, where A is the set of ultrafilters of X containing A.

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Topological structure

The Stone-Čech compactification βX of a (locally compactHausdorff) topological space X is a compact topological spacecontaining X as a subspace such that

βX∃!βf

!!C

CC

C

Xf

//?�

id

OO

K

holds for any continuous map from X to any compact space K.If we associate a set X with the discrete topology, what is βX?

It coincides with U(X), the set of all ultrafilters on X, in whichevery x ∈ X corresponds to the principal ultrafilter Ux. A setV ⊆ βX is open if and only if it is a union of some sets of the formA for A ⊆ X, where A is the set of ultrafilters of X containing A.

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Čech, Stone, Tychnoff

Both Stone 6 and Čech 7 explicitly introduced the concept ofStone-Čech compactification in 1937. But Tychnoff 8 alreadyimplicitly indicated this construction in 1930.

6Marshall Stone, Applications of the theory of Boolean rings togeneral topology, Transactions of the American Mathematical Society, 1937.

7Eduard Čech, On bicompact spaces, Annals of Mathematics, 19378Andrey N. Tychnoff, Über die topologische Erweiterung von Räumen,

Mathematische Annalen, 1930.39 / 47

https://www.ams.org/journals/tran/1937-041-03/S0002-9947-1937-1501905-7/

https://www.jstor.org/stable/1968839

https://eudml.org/doc/159396


Stone

Stone-Čech compactification is an important tool in studying therelationship between a topological space and the ring of allreal-valued continuous functions on it.

For each Boolean algebra, the topological space constructed fromits ultrafilters is known as its Stone space. There is the so-calledStone duality between the category of Boolean algebras andhomomorphisms and the category of Stone spaces and continuousmaps, which is related to Gelfand-Naimark duality.

As a stormiest incident in a stormy period, Marshall Stonerecruited S.S. Chern (陈省身) to the University of Chicago in 1949.

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https://ncatlab.org/nlab/show/Stone+duality

https://www.researchgate.net/publication/26535913_Gelfand_Theorem_Implies_Stone_Representation_Theorem_of_Boolean_Rings

http://people.maths.ox.ac.uk/hap/GuramBezSlides.pdf

http://www-history.mcs.st-and.ac.uk/Biographies/Stone.html


Algebraic structure

If X has in addition a semigroup structure, the multiplication in Xcan be naturally extended to βX。Just use twice the extensionproperty described in the following picture:

βX∃!βf

!!DD

DD

Xf

//?�

id

OO

βX

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Combinatorial largeness

IP: Idempotent; infinite-dimensional parallelepipedFor any commutative semigroup X, a set A ⊆ X is an IP-set(combinatorially large set) provided A ⊇ FS(Y ) for an infinite setY ⊆ X.

• A set A is an IP set if and only if it is contained in anidempotent ultrafilter in the semigroup βX!

• By a theorem of Ellis (1958), every compact left-topologicalsemigroup has an idempotent element.

• Combining the above two facts, Hindman’s Theorem (1974)follows!

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https://math.umn.edu/directory/robert-ellis

https://homepage.univie.ac.at/david.fernandez-breton/slides-blast.pdf


Sum-product

If N is partitioned into a finite number of parts, then we can alwaysfind two distinct positive integers a and b such that a + b and abfall into the same part. 9

9Joel Moreira, Monochromatic sums and products in N, Annals ofMathematics 185 (2017) 1069–1090.

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Combinatorial structure

The Buneman graph of a finite split system is studied a lot inphylogenetic combinatorics. What is the structural limit 10 of allsuch finite Buneman graphs? What is the global geometry of the‘Buneman graph’ of βX?

10J. Nešetřil, P.O. de Mendez, A model theory approach to structural limits,Comment. Math. Univ. Carolin. 53 (2012) 581–603.

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Further readings

• A. Blass, Ultrafilters: where topological dynamics = algebra =combinatorics, 1993.

• Terence Tao, Higher Order Fourier Analysis (高阶傅里叶分析), Higher Education Press, 2016.

• A.M. Vershik, The theory of filtrations of subalgebras,standardness, and independence, Russian Math. Surveys 72(2017) 257–333.

• Evan Warner, Kiddie talk: Ultraproducts and Szemerédi’Theorem, Stanford graduate student seminar, 2012.

• Evan Warner, Ultraproducts and the foundations of higherorder Fourier analysis, Undergraduate thesis, PrincetonUniversity, 2012.

• H.-G. Zirnstein, Formulating Szemerédi’s Theorem in terms ofultrafilters, PhD thesis, University of Leipzig, 2012.

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https://arxiv.org/abs/math/9309203

http://www.pdmi.ras.ru/~avershik/

http://www.math.columbia.edu/~warner/KIDDIEtalk.pdf

http://www.math.columbia.edu/~warner/UndergradThesis.pdf

https://www.cs.rutgers.edu/faculty/endre-szemeredi


有涯与无尽

漱溟自述文录

· · ·

我生有涯愿无尽

心期填海力移山梁漱溟 (1893–1988)

Without music life would be amistake. – Friedrich WilhelmNietzsche

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Thank you!

Without mathematics life would be a mistake.47 / 47

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