Nonstandard models of the reals and symmetrical completeness

24.11.2022 15:00 - 16:30

F.-V. Kuhlmann (U of Szczecin, PL)

The notion of power series fields provides an easy method for the construction of nonstandard models of the ordered field of real numbers. I will define them, as well as Hahn products, which are their equivalent in the case of ordered abelian groups. The question arises whether these power series models can also have additional structures or properties that we know from the reals. For example, it was shown in joint work with Salma Kuhlmann and Saharon Shelah that they do not admit exponential functions which have the same elementary properties as the exponential function on the reals. In a different direction, the question came up whether they could support generalizations of Banach’s Fixed Point Theorem. I will introduce the notions of symmetrically complete ordered fields, abelian groups and sets and characterize those power series models of the reals that are symmetrically complete. They indeed admit a (nonarchimedean) generalization of Banach’s Fixed Point Theorem. Their construction is the result of joint work with Katarzyna Kuhlmann and Saharon Shelah. It heavily relies on the analysis of cuts in ordered power series fields and Hahn products.

Students at Uni Wien are required to attend in person.




SR 10, 1. Stock, Koling. 14-16, 1090 Wien