# Laudatio

Saharon Shelah was awarded an honorary PhD of the TU Wien, Austria, in 2019-12-03.

Martin Goldstern held the following Laudation:

Liebe Kolleginnen und Kollegen, עמיתים יקרים, dear colleagues!

I am very happy to say a few words about Saharon Shelah, who will be awarded a doctoral degree *honoris causa* of our university. *honoris causa* means that the purpose of this degree is to honor him for his achievements, but I rather consider it an honor for myself and for my university that we have such a close scientific connection.

Today it is exactly 50 years ago that Saharon was waiting for his Ph.D. thesis to be approved by the Hebrew University of Jerusalem. Also tomorrow and yesterday it would be 50 years ago, because in fact it took a few months between the submission of his thesis and the formal acceptance. Anyway, also his first two papers appeared in 1969. (That was a meager year — on average he publishes about 20 papers per year.)

Already in this 1969 thesis he introduced the notion of *stable theories*, a concept that has become central in model theory, and has led him to explore and develop a new area in this field; through work of more than a decade he arrived at his celebrated "main gap" theorem:
omitting some technical details — for any theory *T* (that is, the set of statements which are true in some structure, say the theory of arithmetic, of the natural numbers or of some vector space) either this theory *T* is "very good", which means that there is a rather explicit description of all models of *T* (structures in which all statements of *T* hold), or *T* is "very bad", and there is a quite explicit reason why such a description is not possible.

Model theory was Saharon's first born, and thus is of special importance in his approach to mathematics. But as many of you know from the biblical story of Jacob (Jakob, יעקב), also the second son may play an important role.

As Saharon was advancing his model theoretic program, it turned out that he needed set-theoretic tools which did not exist yet. So he created these tools, and thus became also responsible for many major advances in set theory.

Like model theory, set theory is a subfield of mathematical logic. The topic of set theory is — to put it modestly — the investigation of different infinities. Or, as I am a set theorist, to put it less modestly: the investigation of THE UNIVERSE of mathematics, because in the 20th century set theory has become the grand unified language of all of mathematics.

From Shelah's achievements in the area of set theory I will just mention *Proper Forcing* and *Cardinal Arithmetic* — these are titles of books by Shelah; the material from these books has had a significant impact on the set theory of the last 3 decades. Some of you know what I am talking about, but time is too short to explain it to the non-set-theorists (it would take perhaps a semester or two to explain it), so I will say nothing about it, except that I will mention a favorite result of Saharon: It has been known that Cantor's famous *Continuum Hypothesis* about the cardinality of the set of real numbers is neither provable nor refutable. But Shelah's theory of cardinal arithmetic led him to a variant that he calls the revised version of the (Generalized) Continuum Hypothesis, and this variant is indeed provable.

Saharon has more than 270 coauthors from all over the world, not only from Israel and North America, but also from South America, from Iran, India, Singapore, China, Japan, even New Zealand, also one from Africa (but so far no coauthor from Antarctica). Many many coauthors hail from Europe, both EU and non-EU, and about 15 coauthors are either in Vienna, or at least have held here a position at one of Vienna's universities, often financed by the Austrian Science foundation. Several coauthors are students, or (like me, 30 years ago) were students when they started working with Saharon, and they all have had the opportunity to learn from him. This includes several students at TU Wien.

Shelah has published about 330 single-author papers, and about 700 papers with coauthors, and many more are somewhere in the pipeline between the first idea and publication by a journal. (Or: between a first idea and the wastepaper basket, which is also a tool of the mathematician.) Now there are people who write lots of shallow papers that contain only trivial variations of earlier papers. From my experience with those 2 dozen of Saharon's papers where I contributed something, Shelah's papers are always substantial and deep. And often difficult. But sometimes he surprises everybody by finding a proof that is amazingly simple, and had just been overlooked by everybody.

As a reviewer wrote about of one of Shelah's first papers: "The proofs are energetic; some of them are combinatorially very clever".

As many of you know, mathematical research is a bit like a journey of exploration to a foreign country. You start in the wilderness, without a map. Sometimes there are roads, or your predecessors have left you even a car, but sometimes you are on foot. You try to climb a hill to get a better view, to see where the next city is, or the closest river.

Saharon brings a lot of tools to these journeys — since many of them were invented by him, he knows not only how to use them, but also how to sharpen or improve them. And I have a suspicion that sometimes he already knows exactly what to expect; instead of climbing a mountain, he has a balloon in his backpack.

I have occasionally witnessed first hand how he works. Many years ago I asked him a question - the specifics are not important now, the question was of the form "is X true?". He showed me a construction that was supposed to prove X, and it was my job to check it and work out the details. After a day or so I came back and told him that I found a gap - this particular construction does not work. Aha!, he said, but if this does not prove X, then nothing will. And immediately he produced a proof that the negation of X must be true.

I was impressed. What I do when I want to understand a mathematical concept is to first analyze examples and counterexamples, and then I hopefully find the right theorem and its proof. But for Saharon, even a wrong proof will give him insight.

Let me conclude by giving a small glimpse of Saharon's philosophy. I once asked him if he has a platonist view of mathematics - what I meant was: in the naive sense that mathematical objects exist in a world of ideas. He replied that he is neither a platonist nor a formalist, certainly not a finitist, but rather a hedonist - math is fun!

Saharon, I wish you many more years of fun,

עד מאה ועשרים שנה ,

לא תכהה עינך ולא ינוס ליחך

May you live to the age of 120, keeping your strength!

TU Wien issued a press release (in German), which includes a photograph of the award ceremony.