Lectures on Algebraic Model Theory by Bradd Hart and Matthew Valeriote

By Bradd Hart and Matthew Valeriote

In recent times, version concept has had impressive good fortune in fixing vital difficulties in addition to in laying off new mild on our realizing of them. the 3 lectures gathered right here current contemporary advancements in 3 such parts: Anand Pillay on differential fields, Patrick Speissegger on o-minimality and Matthias Clasen and Matthew Valeriote on tame congruence concept.

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Lectures on Algebraic Model Theory

In recent times, version conception has had notable good fortune in fixing vital difficulties in addition to in laying off new gentle on our realizing of them. the 3 lectures amassed right here current fresh advancements in 3 such parts: Anand Pillay on differential fields, Patrick Speissegger on o-minimality and Matthias Clasen and Matthew Valeriote on tame congruence conception.

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Aber nehmen wir f¨ ur einen Augenblick an, die Schulleitung hat ihre Einteilung der Sch¨ uler vorgenommen und f¨ ur jede Schulklasse eine Liste mit den Namen der Sch¨ uler erstellt, die zu dieser Schulklasse geh¨oren sollen. Nehmen wir ferner an, die Schulleitung hat noch nicht u uft, ob jeder ¨berpr¨ Sch¨ uler in genau einer Schulklasse eingeteilt ist. Dann behaupte ich, wenn man in den drei Aussagen 1)-3) die Sch¨ uler Alfred, Ben und Christoph durch beliebige Sch¨ uler ersetzt und die Aussagen richtig sind f¨ ur jede Kombination der Sch¨ ulernamen, dann ist sichergestellt, daß auch jeder Sch¨ uler in genau einer Schulklasse eingeteilt ist.

Wir nennen die Menge P(M) := {A | A ⊆ M} aller Teilmengen von M die Potenzmenge von M. 9 P(∅) = {∅}, P({1}) = {∅, {1}}, P({1, 2}) = {∅, {1}, {2}, {1, 2}}. 10 (Potenzmengen endlicher Mengen) Sei M eine endliche Menge mit n = |M|, so ist |P(M)| = 2n . Beweis: Wir f¨ uhren den Beweis durch Induktion nach n. ¨ § 5. MACHTIGKEIT VON MENGEN 31 Induktionsanfang: n = 0: Dann ist M = ∅ und P(M) = {∅} hat genau 1 = 20 Elemente. Induktionsschritt: n → n + 1: Sei also |M| = n + 1. Wir w¨ahlen ein y ∈ M und setzen N = M \ {y}, so daß |N| = |M| − 1 = n.

Ist m ≤ n, so definiere f : M → N durch f(xi ) = yi f¨ ur i = 1, . . , m. Dann gilt f¨ ur i, j ∈ {1, . . , m} mit i = j f(xi ) = yi = yj = f(xj ). Mithin ist f injektiv. Ist umgekehrt f : M → N eine injektive Abbildung, so gilt f(M) = {f(x1 ), . . , f(xm )} ⊆ N eine Teilmenge von paarweise verschiedenen Elementen. Mithin enth¨alt N mindestens m Elemente, und folglich gilt m ≤ n. b. Ist m ≥ n, so definiere f : M → N durch f(xi ) = yi f¨ ur i = 1, . . , n und f(xi ) = y1 f¨ ur i = n + 1, . .

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