LECTURE: M-W-F: 1-1:50 @ RH 440R

INSTRUCTOR: NAM TRANG
OFFICE HOURS: M 10-12, W 2-3 @ RH 410N

POLICIES AND COURSE GRADES:     The final grades will consist of: 50% HW + 50% Final. I plan to give about 5 to 6 homework assignments. The final will take place W Dec 9th 1:30-3:30.

HOMEWORK ASSIGNMENTS:     HW1    HW2    HW3    HW4    HW5    HW6

COURSE PROGRESS
Week 0
F: Axioms of ZFC
Week 1
M: Remarks regarding ZFC axioms, basic properties of well-ordered sets, order-preserving functions from a well-ordered set to another, theorem on comparability of two well-ordered sets
W: Defining ordinals, proving the \epsilon relation well-orders the ordinals
F: Transfinite induction, recursion, define ordinal arithmetic
Week 2
M: Ordinal arithmetic (cont.), constructive definition of V, ranks for sets
W: Well-founded relations, ranks, induction/recursion on well-founded, set-like relations, transitive collapse maps
F: Well-founded, set-like, extensional relations (cont.), example of a well-founded, set-like relation that is not extensional on some set P, proof of Mostowski collapse theorem.
Week 3
M: Cardinality, cardinal, cardinal arithmetic
W: Cardinal arithmetic (cont.), successor cardinals, limit cardinals, Godel's order
F: Cardinal arithmetic (cont.), proof that \kappa.\kappa = \kappa for infinite cardinal \kappa, closure of a set under a family of finitary functions
Week 4
M: Closure of a set under a family of finitary functions (cont.), reprove \kappa.\kappa=\kappa, introduce the notion of cofinality
W: Cardinal arithmetic (cont.), inaccessible/strong limit cardinals, Koenig's lemma and applications
F: Gimel function and relations to the continuum function
Week 5
M: Gimel function and relations to the continuum function (cont.), introduce the hierarchy H_\kappa
W: H_\kappa (cont.)
F: Club and stationary sets, closure of club sets under intersections
Week 6
M: Proof that the club filter C_\kappa is normal, \kappa-complete
W: Proof of Fodor's lemma, Solovay theorem on splitting stationary sets
F: Solovay theorem on splitting stationary sets (cont.), proof of Kunen's inconsistency theorem
Week 7
M: Jech's definition of stationarity, characterize clubness in terms of closure points of a function
W: Holiday
F: Jech's definition of stationarity (cont.), Woodin's definition of stationarity.
Week 8
M: Woodin's definition of stationarity (cont.), Introducing first order logic with the goal of proving the completeness/compactness theorems.
W: Define L-proofs/provability, axioms and rules of inference, proof of the soundness theorem
F: Proof of the soundness theorem (cont.), define and characterize deductive (in)consistent sets, state the completeness theorem, and model existence theorem.
Week 8
M: Woodin's definition of stationarity (cont.), Introducing first order logic with the goal of proving the completeness/compactness theorems.
W: Define L-proofs/provability, axioms and rules of inference, proof of the soundness theorem
F: State completeness/compactness theorem, state the model existence theorem, equivalent definitions of deductively consistent sets of sentences
Week 9
M: Prove the deduction theorem, prove the completeness theorem from model existence theorem, prove the compactness theorem.
W: Start the proof of Henkin Model theorem.
F: Thanksgiving, no class.
Week 10
M: Finish the proof of Henkin Model theorem.
W: Finish the proof of Henkin extension theorem and the proof of model existence theorem.
F: Applications of compactness theorem: upward Lowenheim-Skolem theorem, and construction of a nonstandard model of the reals.