Verallgemeinerte Quantorenelemination und was nun?
Wednesday, 9.5.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Wir charakterisieren Klassen von (endlichen und unendlichen) Strukturen, die eine verallgemeinerte Quantorenelimination erlauben. Dabei erlaubt die Klasse K verallgemeinerte Quantorenelimination, wenn jede in der Logik der ersten Stufe definierbare Eigenschaft in K bereits durch eine Anzahl q von Quantoren ausgedrückt werden kann, die nur von K abhängt.\nFalls q = 0 gewählt werden kann, erhalten wir somit den \nklassischen Begriff der Quantorenelimination.\n
The Paris-Harrington theorem
Wednesday, 16.5.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Let IRT, the infinite form of Ramsey's theorem, be the statement that for \(c\) a finite non-empty set and \n\(\bpi:[\bomega]^k \blongrightarrow c\) there is an infinite \(Y \bsubset \bomega\) with \(\bpi\) constant on \([Y]^k\). \nSuch an \(Y\) is called homogeneous for \(\bpi\). \n\nLet FRT, the finite form of Ramsey's theorem, be the statement that for \(c\) a finite non-empty set and \nevery positive \(m\) and \(k\) in \(\bomega\) there is an \(n \bin \bomega\) such that whenever \(X\) is a set of size \(n\) \nand \(\bpi: [X]^k \blongrightarrow c\) there is a set \(Y \bin [X]^m\) which is homogeneous for \(\bpi\). \n\nA (finite) subset \(Z\) of \(\bomega\) is called large if the size of \(Z\) is at least \(\bmin Z\). \n\nThe Paris-Harrington statement, PH, is FRT with the strengthened conclusion that the homogenous set \(Y\) \nmay be taken to be large. \n\n\n\nFRT may be deduced from IRT, as can PH; FRT may be proved in Peano arithmetic PA, thus without using the axiom \nof infinity; the remarkable result (1972, published 1977 in the Handbook of Mathematical Logic) of \nParis and Harrington is that PH is too strong to be provable in PA. \n\nThis talk will expound the work of many authors to show that PH fails in many non-standard models of PA. \n
Iterated Ultrapowers in Set Theory
Wednesday, 30.5.18, 16:05-17:05, Raum 404, Ernst-Zermelo-Str. 1
Given a measurable cardinal in an inner model of set theory we can\nconstruct its ultrapower, which is smaller than any of its factors.\nFollowing Kunen's work, we explain this process and its iteration.\n