Modelle der Mengenlehre Exam

Transcription

1 Prof. Dr. Hans-Dieter Donder Summer Term 2015 Iosif Petrakis Modelle der Mengenlehre Exam Family Name: Student ID: First name: Term: Degree course: Bachelor, PO Master, PO Lehramt Gymnasium: modularisiert nicht modularisiert Diplom Other: Major subject: Mathematik Wirtschaftsm. Inf. Phys. Stat. Minor subject: Mathematik Wirtschaftsm. Inf. Phys. Stat. Credit Points to be used for Hauptfach Nebenfach (Bachelor / Master) Please switch off your mobile phone and do not place it on the table; place your identity and student ID cards on the table so that they are clearly visible. Please check that you have received all ten problems. Please do not write with the colors red or green. Write on every page your family name and your first name. Please make sure to submit only one solution for each problem; cross out everything that should not be graded. If you need a tutorial certificate (Übungsschein / nicht modularisiert) please fill in the form on the second next page. By entering a pseudonym (e.g. the last four digits of your student ID number) in the appropriate box on the left at the top of the next page you will give your permission to the publication of your results on the lecture s homepage. You have 120 minutes in total to complete this examination. Good luck!

2 Pseudonym /4 /4 /4 /4 / /4 /4 /4 /4 /4 /40

3 Please read carefully each of the following statements, decide whether it is true or false and tick your answer accordingly. Each correct answer gives one point. Each false answer gives zero points. The optimal total sum is 40 points. Exercise 1: a. The axiom scheme of Replacement is the following scheme x v y (y v z (z x φ(z, y, w)). b. If F is a function and if F A / V, then A / V. c. If n > 0, then ZF x 0, x 1,..., x n ( n i=1 (x i 1 x i ) (x n x 0 )). d. The intersection of two proper classes is a proper class. Exercise 2: a. If u u, then u is transitive. b. A set is an ordinal number if and only if it is a transitive set of transitive sets. c. There is no bijection between the open unit interval (0, 1) and R that takes rationals to rationals and irrationals to irrationals. d. Assume that a is a finite set and f : a a. Then f is an injection if and only if rng(f) = a.

4 Exercise 3: a. There is an ordinal α such that V α / V. b. If F : On On is increasing i.e., α,β On (α < β F (α) < F (β)), then F (λ) = α<λ F (α), for every limit ordinal λ. c. If F : On On satisfies the property F (λ) = α<λ F (α), for every limit ordinal λ, then F is increasing. d. If F is increasing, then α,β On (F (α) < F (β) α < β). Exercise 4: a. If F is a function such that n V ω+n, then F ω V ω 2. b. The limit ordinals form a closed and unbounded class. c. x is a successor ordinal is not equivalent to a Σ 0 -formula. d. x is a limit ordinal is equivalent to a Σ 0 -formula. Exercise 5: a. cf(ω + ω) = ω. b. cf(ω 2 ) > ω. c. ω α+1 is regular, for every ordinal α. d. If λ < cf(κ) and f : λ κ, then rng(f) is bounded in κ.

5 Exercise 6: a. V = L AC + GCH. b. L is an inner model of ZF. c. If V = HOD, then V = OD. d. If a = {2n n ω}, then a OD. Exercise 7: a. L ω V ω. b. Def(ω) = P(ω). c. If α > ω is a limit ordinal, then L α = ZF. d. V is not an inner model of ZF. Exercise 8: W is an inner model of ZF. a. If ZF φ, then ZF φ W. b. If ZF φ W and ZF is consistent, then ZF + φ is consistent. c. If φ( x) is Π 1, then x W (φ( x) φ W ( x)). d. If φ( x) is Σ 1, then x W (φ( x) φ W ( x)).

6 Exercise 9: Let M be a countable transitive model of ZFC, <P,, 1> M a set of conditions, G is M-generic, and the corresponding forcing relation. a. p xφ if and only if a M P p φ(a). b. p φ ψ if and only if q p r q(r φ r ψ). c. If E P and E M, then G E or g G e E(e, g are incompatible). d. Let P = {p M p : n ω, n ω}, <P,, > is the conditions set, G P is M-generic and f = G : ω ω. Then for every g M there exists n ω such that f(n) > g(n). Exercise 10: Suppose that M is a countable transitive model for ZFC, < P, >, where P = {p : n {0, 1, 2, 3} n ω}, G P is M-generic and Φ is a name for G. a. dom(φ) = ω. b. ˆ3 rng(φ). c. There is some p P such that p (Φ takes the value ˆ3 exactly 3 times). d. Φ / L.