# New Stuff from Jeremy Avigad

Jeremy posted this to FOM yesterday:

I’d like to announce a review I have written of two books that deal with logic and foundations in the early twentieth century: Calixto Badesa’s The Birth of Model Theory and Dennis Hesseling’s Gnomes in the Fog. The review, which will appear in the Mathematical Intelligencer, can be found on my web page under “Reviews.”

While I am at it, I’d like to mention two other papers, still in preprint form, that may be of interest to the FOM audience. Both can be found on my web page, under “Research.”

The first, “A formally verified proof of the prime number theorem,” describes a formalization of the PNT using a mechanized proof assistant, Isabelle. The formalization was joint work with Kevin Donnelly, David Gray, and Paul Raff.

The second, “Quantifier elimination for the reals with a predicate for the powers of two,” [with Yimu Yin] is a syntactic proof of a result due to van den Dries, namely, that the theory of the reals as an ordered field with a predicate for the powers of two is decidable. This result is interesting in that it subsumes two of the most important decidability results of the last century: the decidability of real-closed fields, due to Tarski, and the decidability of the additive theory of the integers, due to Presburger. It turned out to be rather difficult to extract an explicit quantifier-elimination procedure from van den Dries’s proof; the best we could do is an algorithm that runs in iterated-iterated-exponential time!

In an MS thesis that he is close to finishing, Yin has also determined a novel equivalence between two q.e. tests which seem to be strictly stronger than q.e. For fun, he has also extended the decidability results to the reals with a predicate for the Fibonacci numbers, and predicates for other sequences defined by appropriate recurrence relations. I will add a link to my web page when he has posted it online.

[Reposted with permission]