1.1. The Erdős Prize in Mathematics.

This prize was first awarded in 1977 and was, at that time, known as the Erdős Prize. The name was changed in 1996 to the Anna and Lajos Erdős Prize reflecting the original wish of Paul Erdős. Anna and Lajos Erdős were Paul Erdős's parents. The prize is awarded to an Israeli mathematician in an institution of higher education in Israel, in the fields of theoretical mathematics or computer science, who was under the age of 41 as of 31 May in the award year. The chairman of the jury making the award is appointed by the Committee of the Israel Mathematical Union. The prize is awarded at the Union's annual conference and the recipient is invited to lecture on their work to that conference.

1.2. The 1977 Erdős Prize to Saharon Shelah.

The Israel Mathematical Union awarded the 1977 Erdős Prize to Saharon Shelah (Hebrew University). This was the first time the prize was awarded so he had the additional honour of being the first recipient. He was 32 years of age when he received the prize.

2.1. The Rothschild Prize.

The Rothschild Prize was established in 1959 by Yad Hanadiv, a philanthropic foundation acting on behalf of the Rothschild family in Israel. Prizes are awarded to support, encourage and advance the Sciences and Humanities in Israel, and recognise original and outstanding published work in the following disciplines: Agriculture, Chemical Sciences, Engineering, Humanities, Jewish Studies, Life Sciences, Mathematics, Physical Sciences and Social Sciences. Prizes are awarded in two-year cycles; in each discipline, a Prize is awarded once in four years.

2.2. Michael Rabin on the Mathematics Rothschild Prize.

Israel is a mathematical and computer science empire. Despite its small size, the State of Israel is recognised world-wide as an important centre for modern mathematical innovation of the highest calibre. In computer science, Israel is second only to the United States as a source of ground-breaking scientific work. This excellence is testified to by invitations of Israeli scientists to deliver keynote lectures at the most important scientific conferences, by their election to leading academies of sciences such as the US National Academy of Sciences, the American Academy of Arts and Sciences, the French Academy of Sciences and the Royal Society, and by being awarded major international prizes. For example, three Israeli scientists were recipients of the A.M. Turing Award, a prize widely considered to be the equivalent, in the field of computer science, of the Nobel Prizes.

The roots of Israeli prominence in mathematics go back to the years before the establishment of the State of Israel. The Hebrew University Einstein Institute of Mathematics was by founded by mathematicians such as B Amira, A A Fraenkel, M Fekete, J Levitzki and T Motzkin. They brought the European tradition of mathematics from the great centres in Germany and Hungary to the tiny university in Jerusalem. They educated a series of brilliant students, many of whom went on abroad to obtain doctorates and later returned to Israel to start a lineage of students and students of students who formed the core of leadership of mathematical research in Israel. These people were joined by a large number of brilliant researchers who emigrated to Israel over the years, bringing with them further diversity and excellence in mathematical research.

The list of Rothschild Prize recipients reflects the strengths and superior achievements of Israel in mathematics and computer science. It is comprised of absolute world class leaders in the study of partial differential equations, applied mathematics, abstract algebra and ring theory, dynamical systems and ergodic theory, theoretical computer science and its applications to cryptography, group theory and group representations and their applications to fields such as combinatorics and physics, mathematical logic, model theory and its profound applications to algebra. Many of the results bear the names of their prize winning innovators and are of fundamental and lasting importance in their respective fields. Some of this work has spawned practical industrial applications of considerable economic value.

Yad Hanadiv acted wisely and with foresight in recognising the importance and high level of Israeli mathematics and in including this subject amongst the Rothschild Prize categories. This inclusion and the roster of outstanding awardees were based on the excellence of Israeli mathematicians and in turn contributed to further foster and enhance this excellence. All in all, the Rothschild Prizes in mathematics represent a brilliant past and present, and a glorious future!

2.3. The 1982 Rothschild Prize to Saharon Shelah.

The 1982 Rothschild Prize was awarded to Professor Saharon Shelah, Mathematics.

3.1. The Karp Prize.

The Karp Prize is awarded by the Association for Symbolic Logic. The Karp Prize, established in 1973 in memory of Professor Carol Karp, is awarded for an outstanding paper or book in the field of symbolic logic. This award is made by the Association on the recommendation of the Association for Symbolic Logic Committee on Prizes and Awards. This prize is given for a "connected body of research, most of which has been completed in the time since the previous prize was awarded." The Prize is awarded every five years and consists of a cash award.

3.2. The 1983 Karp Prize to Saharon Shelah.

The 1983 Carol Karp Prize was awarded to Saharon Shelah of the Hebrew University of Jerusalem, Israel, for his work on the number of nonisomorphic models of first order theories. The award was made by the Association for Symbolic Logic Council at its annual meeting on 7-8 January 1983 in Denver, Colorado.

4.1. The George Pólya Prize.

George Pólya Prize in Applied Combinatorics is awarded by the Society for Industrial and Applied Mathematics (SIAM). It is intended to emphasise applications of combinatorics and is funded by the estate of Stella Pólya in memory of her husband George. The prize is a modification of the older George Pólya Prize in Combinatorics, originally established as the George Pólya Prize in 1969.

The George Pólya Prize in Combinatorics, originally established in 1969, is awarded every four years for a notable application of combinatorial theory. The prize is broadly intended to recognise specific work. The award may occasionally be made for cumulative work, but such awards should be rare.

The prize committee will consist of a panel of five SIAM members appointed by the president. One of the members will be designated by the president as chair. Members of the prize committee must be appointed at least eighteen months before the prize award date.

The prize should be given for a single piece of work. The prize may be awarded to multiple individuals but only for contributions to a single piece of their collaborative work.

4.2. The 1992 George Pólya Prize to Gil Kalai and Saharon Shelah.

The 1992 George Pólya Prize was awarded to Gil Kalai and Saharon Shelah for their work on proving that a closed planar set with the property that each of its visually independent subsets has at most one accumulation point is a countable union of convex sets.

5.1. The 1998 Israel Prize.

The Israel Prize is the most important and prestigious award given in the State of Israel. It was initiated in 1953 by the then Minister of Education, Ben-Zion Dinur, and has been awarded continuously since then.

Each year, the committees of judges submit their recommendations to the Minister of Education for the award of the prize in various and diverse fields of activity and creativity in Israel.

The winners are Israeli citizens - individuals, or in exceptional cases, partners in achievement who have demonstrated special distinction, excellence and breakthrough in their field or who have made a special contribution to Israeli society.

The award ceremony is held annually, on the evening of Independence Day in Jerusalem, the capital of Israel, at an official and state ceremony, and in the presence of the heads of state: the President, the Prime Minister, the Speaker of the Knesset, the President of the Supreme Court, the Mayor of Jerusalem and the Minister of Education - who is the government's representative for the issue of the Israel Prize.

5.2. The 1998 Israel Prize to Saharon Shelah.

The 1998 Israel Prize for mathematics was awarded to Saharon Shelah.

6.1. The János Bolyai International Mathematics Prize.

In 1902, in honour of the 100th anniversary of the birth of the world-famous Hungarian mathematician János Bolyai, the Hungarian Academy of Sciences established the ten thousand crown international award for outstanding mathematical work.

In addition to preserving Bolyai's memory, the original goals of establishing the award also included the ideological replacement of the missing Nobel Prize in Mathematics.

The first laureate in 1905 was the Frenchman Henri Poincaré, one of the most versatile mathematicians of the 19th century, and in 1910 the German David Hilbert received the award. The awarding of the medal was discontinued after the outbreak of World War I.

The Hungarian Academy of Sciences re-established the award in 1994 under the name János Bolyai International Mathematics Prize. The award carries a cash prize of US$25,000 and a gilded bronze medal made using the original designs.

The Bolyai Prize is awarded every five years by the Hungarian Academy of Sciences to the author of the most outstanding, groundbreaking mathematical monograph published in the previous fifteen (previously ten) years, anywhere and in any language, presenting his/her own new results and methods, taking into account the author's previous scientific work.

One year before the award is given, the Academy's Department of Mathematical Sciences elects a committee consisting of five regular members and five outstanding foreign mathematicians and appoints its chairman. The committee reports its decision to the department chairman no later than three months before the award ceremony. The committee elects its speaker from among its members, who will present the award recipient's work in detail and prepare a written report on it when the award ceremony is held. The chairman also votes in the committee and has a casting vote in the event of a tie.

6.2. The 2000 János Bolyai Prize to Saharon Shelah.

Saharon Shelah (Hebrew University, Jerusalem, Israel and Rutgers University, New Jersey) has been awarded the János Bolyai International Mathematical Prize of the Hungarian Academy of Sciences, which consists of a medal and an award of $25,000. The award was presented on 4 November 2000 in a ceremony in Budapest, Hungary.

This award was established in 1903 by the Hungarian Academy of Sciences in honour of János Bolyai, co-discoverer of non-Euclidean geometry, and was presented to H Poincaré in 1905 and to D. Hilbert in 1910, after which various historical events, beginning with the first World War, forced its interruption. The Academy has decided to renew the award, and Professor Shelah is the first recipient in modern times. In keeping with the original plan, the prize will be awarded every five years to the author of the best mathematical monograph containing original research which has been published in the previous ten years.

The monograph for which Professor received the award is his work Cardinal Arithmetic, published by Oxford University Press in 1994, in which he presented his profound and sophisticated pcf theory, a new approach to cardinal arithmetic. This work attests to his view that there is an essential core of set theory unaffected by the famous independence results of the last four decades, in which a great deal of work remains to be done, His applications of pcf theory, many of which are given in his monograph, constitute a breakthrough in many areas of pure, as well as applied set theory.

In his 700-plus papers and several books Professor Shelah has had an unparalleled influence on modern set theory and model theory. His solutions to deep and longstanding problems and his independence techniques forever changed the landscape of set theory; this is even more true in model theory where his concepts and methods have completely revolutionised the area. He also solved a number of famous problems arising in other branches of mathematics, such as algebra, combinatorics, and topology.

6.3. Saharon Shelah's Bolyai Prize lecture.

After he received the International Bolyai Prize in Mathematics on 4 November 2000 in Budapest, Hungary, Saharon Shelah delivered the lecture You Can Enter Cantor's Paradise. Here is the beginning of the lecture:

I will try to use a spiralic presentation returning to the same points on higher levels hence repeating ourselves, so that a reader lost somewhere, will not go away empty handed. Also I will assume essentially no particular knowledge and I will say little on the history to which many great mathematicians contributed.

1. Hilbert's first problem

Recall (Cantor):

• We say that two sets $A, B$ are equinumerous (or equivalent) if there is a one-to-one and onto mapping from $A$ onto $B$;

• The Continuum Hypothesis, CH, is the following statement:

every infinite set of reals is either equinumerous with the set $\mathbb{Q}$ of rational numbers, or is equinumerous with the set $\mathbb{R}$ of all reals;

• For a set $X$, let $\mathcal{P}(X)$ denote its power set, i.e., the set of all subsets of $X$.

The Generalized Continuum Hypothesis, GCH, is the statement asserting that for every infinite set $X$, every subset $Y$ of the power set $\mathcal{P}(X)$ is either equinumerous with a subset of $X$, or is equinumerous with $\mathcal{P}(X)$ itself.

I think this problem is better understood in the context of:

1.1. Cardinal Arithmetic. Recall (Cantor), that we call two sets $A, B$ equivalent (or equinumerous) if there is a one-to-one mapping from $A$ onto $B$; the number of elements of $A$ is the equivalence class of $A$ denoted by $|A|$, we call it also the power or the cardinality of $A$. Having defined infinite numbers, we can naturally ask ourselves what is the natural meaning of the arithmetical operations and the order. There can be little doubt concerning the order:

• $|A| < |B|$ iff $A$ is equivalent to some subset of $B$.

We know that

• any two infinite cardinals are comparable so it is really a linear order

• any cardinal $\lambda$ has a successor $\lambda^{+}$, which means that

• $\lambda < \mu$ ⇔ $\lambda^{+} < \mu$.

Well, but what about the arithmetical operations? There are natural definitions for the basic operations:

• addition (such that $|A \bigcup B| = |A| + |B|$ when $A, B$ are disjoint),

• multiplication (such that $A \times B = |A| \times |B|$),

• exponentiation (such that = $|A|^{|B|} = |^{B}A|$, where $^{B}A$ = {$f$ | $f$ a function from $B$ to $A$}).

A mathematician is allowed to choose his definitions and give them "nice" names, but do those operations have any laws? Are there interesting theorems about them? For the second question it is a big yes but is outside the scope of this article. For the first question, the answer is clear cut: all the usual equalities hold, that is, addition and multiplication satisfy the commutative, associative, and distributive laws and their infinite parallels. Also for exponentiation, e.g. $\lambda^{\mu^{\kappa}} = \lambda^{\mu\times \kappa}, \lambda^{\mu} \times \lambda^{\kappa} = \lambda^{\mu+\kappa}$. However this does not hold for the inequalities. For every infinite cardinal $\lambda$ we have $\lambda = \lambda + 1$. This should not surprise us; it is to be expected that allowing infinite numbers will "cost" us some losses, (as extending $\mathbb{N}$, the integers, to rationals "costs" us the existence of successor and the proof by induction; this is very clear in a posteriory wisdom, of which we all have a lot). Cantor was going around asking: are there more points in the plane than on the line? People answered him: don't you see that there are? But it is false, the line and the plane are equinumerous.

In fact we can totally understand addition and multiplication, as the following very nice rules holds for infinite numbers:

• $\lambda + \mu$ = max$(\lambda, \mu)$

• $\lambda \times \mu$ = max$(\lambda, \mu)$

School children would have loved such arithmetic!

You may wonder: is this not too good? Maybe all infinite numbers are equal so this arithmetic is not so interesting? But Cantor showed that $2^{\lambda} > \lambda$, meaning in particular that there are more reals than natural numbers; noting that he called the number of natural numbers $\aleph _{0}$ this is the first infinite cardinal and showed that the number of reals is $2^{\aleph _{0}}$.

Recall that every infinite number $\lambda$ has a successor, one bigger than it but smaller or equal to any bigger number, and it is denoted by $\lambda^{+}$.

Now mathematicians tend to conjecture that things are nice and well understood. So, having only two natural operations to increase a cardinal, what is more natural than to conjecture that those two operations, $\lambda^{+}$ and $2^{\lambda}$ are equal. Also mathematicians tend to conjecture either that whatever they cannot prove may fail, or whatever they cannot build counterexample to is true; and being unable to construct an intermediate cardinal between $\lambda$ and $2^{\lambda}$ (e.g. $\aleph _{0}$ and $2^{\aleph _{0}}$) it is natural to conjecture that there is nothing between them. This is

1.2. Hilbert's first problem, general version. The "generalised continuum hypothesis", or GCH, says: for every infinite number $\mu$, its power $2^{\mu}$ is its successor $\mu^{+}$.

The interest is that if GCH holds, then not only addition and multiplication are easy, but also exponentiation is easy ...

Dream: Find the laws of (infinite) cardinal exponentiation.

It has been assumed that if we understand cardinal arithmetic, that is (taking for granted the understanding of addition and multiplication) understand the behaviour of exponentiation, we will generally understand set theory much better, and so solve problems from many branches of mathematics in full generality.

7.1. The Wolf Prize in Mathematics.

There is no Nobel prize in mathematics. Perhaps this is a good thing. Nobel prizes create so much public attention that mathematicians would lose their concentration to work. There are several other prizes for mathematicians. There is the Fields Medal (only for mathematicians); it honours outstanding work and encourages further efforts.

Then there is the Wolf Prize. The Wolf Foundation began its activities in 1976. Since 1978, five or six annual prizes have been awarded to outstanding scientists and artists, irrespective of nationality, race, colour, religion, sex or political view, for achievements in the interest of mankind and friendly relations among people. In science, the fields are agriculture, chemistry, mathematics, medicine, and physics; in the arts, the prize rotates annually among music, painting, sculpture and architecture.

The Fields Medal goes to young people, and indeed many mathematicians do their best work in the early years of their life. The Wolf Prize often honours the achievements of a whole life. But it may also honour the work of young people. The first Wolf Prize winners in mathematics were Izrail M Gel'fand and Carl L Siegel (1978). Siegel was born in 1896 and Gel'fand in 1913. Several prize winners were born before 1910. Thus the achievements of the prize winners cover much of the twentieth century.

7.2. The 2001 Wolf Prize in Mathematics to Saharon Shelah.

The Wolf Prize Laureate in Mathematics 2001 is Saharon Shelah (Hebrew University of Jerusalem, Israel:-

... for his many fundamental contributions to mathematical logic and set theory, and their applications within other parts of mathematics.

Professor Saharon Shelah is a leading mathematician in the foundations of mathematics and mathematical logic. His staggering output, of 700 papers and half a dozen monographs, includes the creation of several entirely new theories that changed the course of model theory and modern set theory, as well as providing the tools to settle old problems from many other branches of mathematics, including group theory, topology, measure theory, Banach spaces, and combinatorics.

Shelah created a number of subfields of set theory, most notably the theory of proper forcing and the theory of possible cofinalities, a remarkable refinement of the notion of cardinality, which led to the proofs of definite statements in areas previously considered far beyond the limits of undecidability. Shelah's work on set theoretic algebra and its applications showed that dozens of areas of algebra involve phenomena that are not controlled by universally-recognised axioms of set theory (independence phenomena). In model theory he carried through a monumental program of deep structural analysis known as "stability theory" which now dominates a large part of the field.

7.3. The American Mathematical Society on the 2001 Wolf Prize.

Professor Saharon Shelah has been for many years the leading mathematician in the foundations of mathematics and mathematical logic. His astonishing output - seven hundred articles and a dozen and a half monographs - contains the creation of several completely new theories that changed the course of modern model theory and group theory, and also provided the tools for solving old problems in many other branches of mathematics, including group theory, topology, measure theory, Banach spaces and combinatorics.

Shelah created several sub-branches of set theory, the most prominent of which are the theory of 'proper coercion' and the theory of 'possible finitude', an impressive refinement of the concept of power, which led to the proof of definitive claims in areas previously considered far beyond the limits of undecidability. His work on algebra from the perspective of set theory and its applications showed that dozens of areas of algebra involve phenomena that are not governed by the universally accepted axioms of set theory (phenomena of independence). In model theory, he carried out a monumental program of in-depth structural analysis known as 'stability theory', which today dominates a large part of this field.

8.1. The EMET Prize for Art, Science and Culture.

The EMET Prize is an annual prize given for excellence in academic and professional achievements that have significant influence on society. The Prizes, in a total amount of one million dollars, are sponsored by the A.M.N. Foundation for the Advancement of Science, Art and Culture in Israel, under the auspices of and in cooperation with the Prime Minister of Israel. Prizes are awarded annually in the Exact Sciences, Life Sciences, Social Sciences, Humanities and Judaism, and Art and Culture.

8.2. The 2011 EMET Prize Jury Statement.

The 2011 EMET Prize is awarded to Saharon Shelah. One of the world's preeminent researchers in the area of Mathematical Logic and its applications to various areas of classical mathematics, for his deep understanding and his long-standing and wide-ranging work which has resolved many conjectures and transformed Model Theory.

8.3. The 2011 EMET Prize Jury CV.

Saharon Shelah was born in Israel in 1945. His father was the poet Yonatan Ratosh.

He attended Tel Aviv University and was awarded a B.Sc. Cum Laude in 1964, and afterwards he served in the IDF's Computing Center.

In 1968 he was awarded an M.Sc., also Cum Laude, and two years later he was awarded his Ph.D. Summa Cum Laude from the Hebrew University under the supervision of Prof M O Rabin. His thesis was on Categoricity of Classes of Models.

Since 1971 he has been teaching and carrying out research at the Hebrew University. In his research he discovered deep connections between model theory, set theory and combinatorics and he created many powerful techniques which have also been applied to algebraic geometry and number theory. In set theory he proved a large number of impressive results and developed two totally new subfields: the theory of proper forcing and the theory of possible cofinalities (PCF), and proved the consistency of various forcing axioms. He has published close to 1000 papers and solved many open problems in mathematics.

Among his accomplishments: in model theory he developed an area called classification theory, by which he solved Morley's Problem and proved that Whitehead's Problem is independent of the axioms.

His research awarded him many prizes, among them the Erdös Prize (1977), the Rothschild Prize (1982), the Karp Prize (1983), the Israel Prize (1998), the Bolyai Prize (2000) and the Wolf Prize (2001). In 1988 he became a Member of the Israel Academy of Sciences and Humanities. Three years later he became a Foreign Honorary Member of the American Academy of Arts and Sciences.

8.4. Saharon Shelah wins 2011 EMET Prize.

Professor Shelah's prize is in the area of the exact sciences. Holder of the Professor Abraham Robinson Chair in Mathematical Logic, he is considered a leading expert in mathematical logic and its use in various applications to classical mathematics. His wide-ranging and penetrating work over a period of some 40 years has contributed greatly to a rethinking of model theory.

9.1. The 2013 Leroy P Steele Prize Awards.

The 2013 AMS Leroy P Steele Prizes were presented at the 119th Annual Meeting of the AMS in San Diego, California, in January 2013. The Steele Prizes were awarded to John Guckenheimer and Philip Holmes for Mathematical Exposition, to Saharon Shelah for a Seminal Contribution to Research, and to Yakov Sinai for Lifetime Achievement.

9.2. The 2013 Leroy P Steele Prize Citation for Saharon Shelah.

The 2013 Leroy P Steele Prize for Seminal Contribution to Research is awarded to Saharon Shelah for his book Classification Theory and the Number of Nonisomorphic Models (Studies in Logic and the Foundations of Mathematics, 92, North-Holland Publishing Co., Amsterdam-New York, 1978; 2nd edition, 1990).

Before Shelah's work, the great theorem of pure model theory was Morley's theorem on categoricity. It concerned a class of theories whose uncountable models are completely determined by their cardinality. Shelah visualised a vast extension of the problem to a classification of arbitrary first-order theories, based on the number of models they may have of a given uncountable size. Solving this problem required some twenty years that made model theory into a mature field, completely transforming its aims, methods, and ability to connect to algebra and geometry.

Shelah isolated the class of stable theories, where finitely generated extensions admit, in a certain local sense, finitary descriptions. He was able to show, on the other hand, that any unstable theory has the maximum set-theoretically permissible number of models. All theories of modules are stable. Among the stable theories, he isolated the superstable theories, analogous to Noetherian rings, and again found many models if this condition fails. These were the first two of a series of dividing lines, characterised by a deep theorem on either side. On the stable side, he was able to define a canonical tensor product of extensions of structures and made it into an incisive tool for the decomposition of structures. An arsenal of notions became available to the previously bare-handed model theorist: algebraic closure, canonical bases, imaginary sorts, domination, forking, regular types. These concepts proved useful beyond the stable framework and led to substantial applications when investigated in algebraic settings. The problem of the number of models was solved in the second edition of his monograph, but the ideas of the solution remained central and proved critical for many others. It would be impossible to imagine model theory today without them.

9.3. Saharon Shelah's prize winning book.

Saharon Shelah's book: "Classification theory and the number of non-isomorphic models", won the Leroy P Steele prize in 2013.

The book discusses the following question: When does a class of mathematical structures (for example a class of groups or a class of fields) have a "structure theorem"? When can we fully understand the structure of the class? The main theorem of the book, the so-called "main gap" gives a resolution of this question for classes specified by a countable set of first-order axioms.

The main concept in Shelah's resolution is the notion of a dividing line. Roughly speaking, a dividing line is a property so that both it and its negation have strong consequences. Typically, a class falling on the bad side of the dividing line does not, in a strong sense, have a structure theorem, while a class falling on the good side is quite well-behaved and can be analysed further (typically via more dividing lines).

9.4. From Saharon Shelah's Introduction.

The aim of this book is to represent work of the author on classification and related topics. The author, in a moment of insanity, believed this would be the easiest way to represent his work. There is no point in trying to convince you that I think the book is important; since otherwise I would not have spent my time writing it. ... So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights. ...

9.5. The 2013 Leroy P Steele Prize Biographical Sketch.

Saharon Shelah earned his B.Sc. from Tel Aviv University, his M.Sc. from the Hebrew University under the supervision of Professor H Gaifman, and his Ph.D. from the Hebrew University under the supervision of Professor M. Rabin. He has taught at the Hebrew University and Rutgers University, among others. He is a member of the Israel Academy of Sciences and Humanities and the American Academy of Arts and Sciences.

9.6. Saharon Shelah's response to winning the 2013 Leroy P Steele Prize.

I am grateful for this great honour. While it is great to find full understanding of that for which we have considerable knowledge, I have been attracted to trying to find some order in the darkness; more specifically, finding meaningful dividing lines among general families of structures. This means that there are meaningful things to be said on both sides of the divide: characteristically, understanding the tame ones and giving evidence of being complicated for the chaotic ones. It is expected that this will eventually help in understanding even specific classes and even specific structures. Some others see this as the aim of model theory; not so for me. Still I expect and welcome such applications and interactions. It is a happy day for me that this line of thought has received such honourable recognition. Thank you.

10.1. The Hausdorff Medal.

The Hausdorff Meda is a mathematical prize awarded every two years by the European Set Theory Society The award recognises the work considered to have had the most impact within set theory among all articles published in the previous five years. The award is named after the German mathematician Felix Hausdorff (1868-1942).

10.2. Third Hausdorff Medal 2017 Announcement.

Ladies and gentlemen, dear friends and colleagues!

It is my honour and pleasure to announce the winner of the third Hausdorff medal of the European Set Theory Society. The Hausdorff medal is awarded biennially (i.e. once every second year) for the most influential work in set theory published in the five years preceding the awarding of the medal. The prize committee, that consists of the members of the Board of Trustees of the Society, decided that the third Hausdorff medal is awarded to Maryanthe Malliaris and Saharon Shelah for their work outlined in the paper:

General topology meets model theory, on p and t, Proc. Natl. Acad. Sci. USA 110 (33) (2013), 13300-13305,

and then expounded in the detailed, 60 page long version:

Cofinality spectrum theorems in model theory, set theory, and general topology, J. Amer. Math. Soc. 29 (1) (2016), 237-297.

10.3. Laudation for the 2017 Hausdorff Medal.

Malliaris and Shelah solved two long-standing and fundamental problems:

First, they solved a more than 50 year old set theoretic problem, going back to Rothberger, by showing that the well-known and important cardinal characteristics p and t of the continuum are actually equal.

Secondly, they solved a 40 year old problem in model theory by showing that the maximality in Keislers order is not characterised by the strict order property, but that a weak order property called SOP2 suffices.

Both results follow from a brilliant analysis of definability in ultraproducts of finite linear orders. This analysis is also unique in proving that there are theories more complex than the stable, i.e. minimal theories but less complex than the maximal class in Keisler's order.

To conclude, this important work of Malliaris and Shelah opens the door for significant and fruitful new interactions between set theory and model theory.

11.1. The Rolf Schock Prize in Logic and Philosophy.

The Schock Prize is 400,000 Swedish Krona (roughly $49,000) and is awarded by the Royal Swedish Academy of Sciences. The prize is named for Dr Rolf Schock, who died in 1986, and whose will set out that a portion of his estate be used to fund four prizes in the fields of logic and philosophy, mathematics, the visual arts and music.

Previous winners of the Schock Prize in Logic and Philosophy include Ruth Millikan (2017), Derek Parfit (2014), Hilary Putnam (2011), Thomas Nagel (2008), Jaako Hintikka (2005), Solomon Feferman (2003), Saul Kripke (2001), John Rawls (1999), Dana Scott (1997), Michael Dummettt (1995), and Willard Van Orman Quine (1993).

11.2. The 2018 Rolf Schock Prizes

The Royal Swedish Academy of Sciences, the Royal Academy of Fine Arts and the Royal Swedish Academy of Music have awarded this year's Rolf Schock Prizes to four outstanding individuals.

The prize amounts to 400,000 Swedish krona per prize area, a total of 1.6 million Swedish krona.

"I am particularly delighted to highlight a prize that so clearly unites science and art, entirely in accordance with Rolf Schock's will. This year's Laureates demonstrate the breadth of the work of the awarding academies. They have all been incredibly creative and leading in their fields," says Göran K Hansson, Chair of the Rolf Schock Foundation and Secretary General of the Royal Swedish Academy of Sciences.

11.3. The 2018 Rolf Schock Prize in Logic and Philosophy.

Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University has been awarded the Rolf Schock Prize for 2018 by the Royal Swedish Academy of Sciences. Shelah was awarded the prize in logic and philosophy "for his outstanding contributions to mathematical logic, in particular to model theory, in which his classification of theories in terms of so-called stability properties has fundamentally transformed the field of research of this discipline."

The prize citation for Shelah reads: "Saharon Shelah has made fundamental contributions to mathematical logic, particularly in model theory and set theory. In model theory, Shelah developed classification theory, concerning the classification of first-order theories in terms of properties of their classes of models. The classes of models of so-called stable theories have structural properties that can be characterised in geometrical terms, while the class of models of an 'unstable' theory lacks structure. Most of contemporary research in model theory builds on Shelah's work. Shelah has also made decisive contributions to set theory, including the development of a new variety of the forcing method and remarkable results in cardinal arithmetic, and he has solved deep problems in other areas, such as algebra, algebraic geometry, topology, combinatorics, computer science, and social choice theory. Shelah has had, and still has, an indisputable and exceptional position in mathematical logic, particularly in model theory. He is almost unbelievably productive, with seven books and more than 1,100 articles to date."

Shelah was born in Jerusalem in 1945. He received his PhD from Hebrew University under the direction of Michael O. Rabin. He held positions at Princeton University (1969-1970) and the University of California Los Angeles before joining the faculty at Hebrew University. He is also distinguished visiting professor at Rutgers University. His honours include the Erdős Prize (1977), the Rothschild Prize (1982), the Karp Prize (1983), the George Pólya Prize (1992), the Bolyai Prize (2000), the Wolf Prize (2001), the Steele Prize for Seminal Contribution to Research (2013), and the Hausdorff Medal (with Maryanthe Malliaris, 2017).

11.4. Saharon Shelah's response on the award of the Rolf Schock Prize.

I would like to thank my students, collaborators, contemporary logicians, my teachers Haim Gaifman and Azriel Levy and my advisor Michael Rabin. I have been particularly influenced by Alfred Tarski, Michael Morley and Jerome Keisler.

11.5. Jouko Väänänen's lecture on Saharon Shelah's contributions at the Rolf Schock Prize presentation.

Jouko Väänänen's lecture begins:

I will give a brief overview of Saharon Shelah's work in mathematical logic, especially in model theory. It is a formidable task given the sheer volume and broad spectrum of his work, but I will do my best, focusing on just a few main themes.
In the statement that Shelah presented to the Schock Prize Committee, he wrote:

I would like to thank my students, collaborators, contemporary logicians, my teachers Haim Gaifman and Azriel Levy and my advisor Michael Rabin. I have been particularly influenced by Alfred Tarski, Michael Morley and Jerome Keisler.

As it happens, in 1971 there was a birthday meeting in Berkeley in honour of Alfred Tarski. All of these people were there and gave talks. Shelah's contribution to the Proceedings Volume of this meeting (Henkin et al., 1974) is number 31 in the numbering of his papers, which now already extends to 1150. The papers are on the internet, often referred to by their "Shelah-number". We sometimes tease Saharon by asking him what is in, say, paper 716, and usually he knows it. Somebody once asked about a result and Saharon said it was in paper number 3. Three hundred what, asked the person, realising that Shelah has hundreds of papers, and Saharon answered, no, paper number 3. In fact paper number 3 (Shelah, 1970) is a very influential paper for the development of model theory.

Shelah made essentially three transformative contributions to the field of mathematical logic: stability theory, proper forcing and PCF theory, the first in model theory and the other two in set theory. He started as a model theorist and I think he still considers himself mainly as a model theorist, but he has extended his interest and work to set theory.

1 Model Theory: Stability Theory

Model theory is a branch of mathematics that deals with the relationship between descriptions or "axioms" in the so-called first-order languages (sometimes also in extensions) and the structures that satisfy these descriptions. This is a very general characterisation of model theory going all the way back to Tarski. "First-order language" means that quantifiers "for all" and "exists" range over elements (not subsets) of the domain. An example is provided by the group axioms:

$\forall x \forall y \forall z (x . (y . z) = (x . y) . z)$

$\forall x (x . 1 = x \lor 1 . x = 1)$

$\forall x \exists y (x. y = 1 \lor y . x = 1)$

which talk about the group elements but not about sets of group elements. If I wanted to say that the group is free, I would have to talk about subsets of the group in order to say that there is a free basis. Other examples of first-order axioms are the field axioms, the axioms of order, the (first-order) Peano axioms and the axioms of set theory.

Originally, more than 100 years ago, there was an idea, advocated, for example, by Hilbert (although Hilbert seemed not to make a great distinction between first- and second-order axiomatisations), that mathematical structures can be understood through their axiomatisations. There is a fundamental philosophical question: how is it possible that we understand infinite objects such as real or complex numbers, with the finite means that we have? How can we be certain about the properties of infinitary things? Hilbert's idea (shared by others) was that we write down axioms and the axioms characterise their models completely. It turned out to be not quite so. Skolem, as early as the 1920s, and then Gödel in the 1930s, showed that there are certain limitations to these attempts. In the 1960s model theory developed quite strongly but mainly using set-theoretic methods. The limitations of the extent to which first-order axiomatisations capture mathematical concepts and structures were exposed in very manifest ways.

When Shelah entered the model theory scene he isolated an instability phenomenon in certain first-order theories. It is something that people like Hilbert, Skolem and Gödel, who came to logic earlier, did not consider and had no idea about. It transforms model theory from the set-theoretic approach into a more geometric and algebraic form. In the 1960s when set-theoretic model theory had become quite complicated, the more geometric approach brought new hope for understanding models of first-order theories by building on the long history of geometry and algebra. In this respect we can think of the weakness of first-order logic, revealed by Skolem and Gödel, as a strength in the hands of modern model theorists, e.g., Shelah. First-order descriptions of structures are at the same time sufficiently strict, keeping the structure from being "too general", and sufficiently tolerant to allow a rich theory and interesting constructions. First-order logic (i.e., language) strikes a kind of very successful balance.

In 1978 Shelah's book Classification Theory (Shelah, 1978) appeared. This is a fundamental book that everybody in model theory rushed to read. It was not an easy book to read but it contained everything that was needed at that time and long after. In particular, it contained the basics of stability theory. Stability theory is now the accepted state-of-the-art and the focus of research for all those working in model theory.

In the June 1982 issue of the Abstracts of the American Mathematical Society Shelah published a paper with the title "Why Am I So Happy?" He had made a landmark breakthrough leading to the so-called Main Gap Theorem (discussed later in this article). Hodges (1987, p. 209) writes:

He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic.

I will now explain what this is about. ...

12.1. The "Ettore Casari" Prize for Logic.

The Italian Society for Logic and the Philosophy of Science (SILFS) established the biennial logic prize named 'Ettore Casari Logic Prize', which is aimed at promoting logical research in Italy. Scholars affiliated to any University or Research Centre based in Italy, irrespective of their role, nationality, gender, or age, are eligible for participation. This prize is part of the UNILOG project "A Prize of Logic in Every Country".

Applicants are requested to submit an original manuscript written in English, not exceeding 30 pages, on any topic that can be considered as pertaining to logic (by the standards of the international community of logicians).

The prize consists in: a) the publication of the selected manuscript in the journal Logica Universalis, b) the payment of the participation costs (travel, accommodation, and registration fee) for the World Congress of Universal Logic, and c) the award of the medal 'Ettore Casari for Logic'.

12.2. The 2024 "Ettore Casari" Prize for Logic to Gianluca Paolini and Saharon Shelah.

The Committee appointed by the Italian Society for Logic and the Philosophy of Science to assign the "Ettore Casari" Prize for Logic 2024 convened in three sessions to discuss and evaluate the eleven submitted works. The evaluation was conducted based on the following criteria: originality, scientific quality, rigour, and clarity of the text. Each work was evaluated according to these established criteria and considered in the context of its scientific relevance and overall quality.

At the conclusion of the evaluations, the Committee unanimously decided to award the "Ettore Casari" Prize for Logic 2024 to the work titled "Torsion-free abelian groups are Borel complete", authored by Gianluca Paolini and Saharon Shelah.

12.3. Award Motivation for the 2024 "Ettore Casari" Prize.

The article addresses and resolves a long-standing problem in descriptive set theory, proving that the Borel space of torsion-free abelian groups with domain ω is Borel complete. It stands out for its theoretical rigour, the depth of its analysis, and the use of techniques which, combined with the achieved result, open promising avenues for research in a relevant area of mathematical logic. Furthermore, the clarity with which highly complex topics are presented highlights and further enhances the scientific excellence of this contribution.

12.4. The Committee for the 2024 "Ettore Casari" Prize.

Agata Ciabattoni (Technische Universität, Vienna), Pierluigi Minari (University of Florence), Silvio Ghilardi (University of Milan), Roberto Giuntini (University of Cagliari), Mario Piazza (Scuola Normale Superiore, Pisa), Francesca Poggiolesi (Université Paris 1 Panthéon-Sorbonne), Giuseppe Primiero (University of Milan).