Interpretable Fields in algebraically closed fields
Wednesday, 19.7.17, 16:30-17:30, Raum 404, Eckerstr. 1
Abstract: D. Marker and A. Pillay proved that in a reduct of an algebraically closed field F, which is non-locally modular and expanding the additive structure, an infinite field is interpretable and then the multiplication on F is definable in this reduct. In their work, they use a result of B. Poizat, which states an infinite field K which is definable in the pure algebraically closed field F is definably\nisomorphic to F. I will present this result and its proof.\n
Essentially Different Functions
Wednesday, 26.7.17, 16:30-17:30, Raum 404, Eckerstr. 1
The terminology "Wesentlich verschiedene Abbildungen" (which means "essentially different functions") is taken from Hausdorff's work "Über zwei Sätze von Fichtenholz\nund Kantorovich'' (1935).\n\nWe will follow Hausdorff's proof of the existence of continuum many essentially different functions: i.e. there is some \(H \bsubseteq {^\bomega \bomega}\) of size continuum\nsuch that for every finitely many \(f_0, \bdots, f_i \bin F\) there is a level \(x \bin \bomega\) such that \(f_l(x) \bneq f_j(x)\) for \(l<j \bleq i\).\n\nWe will then see how to generalize the result to find a family of size continuum of "independent functions" using a construction with trees. If the audience is\ninterested, we could also compare it with some other well known (but less pictorial) proofs.\nIf time remains, we will show how the existence of continuum many independent functions applies to prove that a finite support iteration of σ-centred forcing notions is\nagain σ-centred (this is a question asked by Goldstern and answered by Blass in Mathoverflow).\n\n