Yet another Sheep

Saharon Shelah is the most productive logician of our time. Shelah has authored and co-authored more than nine-hundred papers in mathematical logic and related areas. Many of these were extremely influential, such as his work on cardinal arithmetic. His first major contribution to economics was of a rather indirect nature: He was one of the reasons why Ariel Rubinstein entered economics. As Rubinstein writes:

"And, more than anything, the subjective nature of the things I will say is exemplified by the person I believe to be most responsible for the fact that I found myself involved in microeconomic theory. Indeed, from the gallery of esteemed teachers I mentioned, he is completely unknown in the field of economics. I doubt he will ever be appointed as an economic advisor to the finance minister and, as far as I know, he is not trained in the field of law. He has not even written articles on mathematical economics and I doubt he intended to lead me to choose the path I did (though he certainly thought I was not talented enough to become involved in his line of work). The man I’m referring to is the mathematical logician Saharon Shelah.

I would sit with my friends at the cafeteria of the National Library in the middle of the day, with our notebooks full of the puzzling things that Saharon Shelah (like other teachers in the Mathematics Department in Jerusalem) wrote on the board. We were in great awe from this encounter with intellectual perfection, but we also had a vague feeling that these things, despite - or perhaps because of - their great abstractness, had some connection to life. We were witness to a strict adherence to norms of definition and proof, and unwillingness to compromise on a single detail. We became acquainted with the beauty and splendor of a model, a statement and a proof, and with the purity of things emanating from the mind.

But these things spoke to us because of our interest in the world around us. We tried to give mathematical concepts some realistic meaning derived from the concepts of our daily world. And we tried to explain these wonderful mathematical statements not only as links between concepts in the mathematic world, but as links in a world of concepts that so fascinated us as young students after the army: the world of human interaction."

What changed Rubstein's fate was being introduced to Amartya Sen's wonderful book Collective choice and social welfare. which brings us back to social choice theory. Gil Kalai, an expert in combinatorics with a blog of his own, introduced Shelah to the problem of generalizing Arrows theorem to arbitrary choice functions (that do not have to come from some rational preference ordering), that satisfy symmetries like neutrality. The result is On the Arrow Property. A paper with one central result and a really, really, really long and complicated proof. The paper is written in the most ideosyncratic style I have ever seen. Just see for yourself. Do it.

Now if anyone asks for nontrivial mathematics in economic theory, here it is.