Axiomatic Set Theory 2008/2009. by Andrey Bovykin
September 2008 - January 2009
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Exam date: April 9, 2009.
Grand Revision Day: April 2, 2009, at 14:00 in SM3.
Revision list for the exam.
All proofs that appeared in the course might appear in the exam (even if the word "proof" is not written in the list below.)
Basics:
Axioms of ZFC, class terms, basic structure of a model of ZFC: ordinals and V_alpha, rank of a set.
Transitive sets, transitive closure, epsilon-induction, ordinal induction.
Relativisation of a formula to a class, checking extensionality, pair, union, infinity in transitive classes, checking ZF^- in transitive classes,
checking the power-set axiom in transitive classes.
Collapsing lemma
Properties of H_kappa, H_kappa satisfies ZF^-, checking that the size of H_{kappa^+} is 2^kappa.
Cardinals
Cofinality, regular and singular cardinals, knowledge of Konig's theorem and the whole story of GCH
successor and limit cardinals
Some logic
Consistency statements, Godel's completeness theorem, Lowengheim-Skolem theorems
stongly inaccessible cardinals, chacacterisation of strongly inaccessible cardinals, V_kappa satisfies ZFC
Proof that ZFC doesn't prove that there is a strongly inaccessible cardinal
Proof that Con(ZFC) does not prove Con(ZFC+ there is a strongly inaccessible cardinal)
Combinatorics
Clubs and stationary sets, sets containing a club comprise a filter, how many clus are there? how many stationary sets are there?
Filters and ultrafilters, existence or nonexistence of nonprincipal ultrafilters on omega (assuming AC), how about without AC?
Diamond, diamond implies CH.
Trees
Construction of an Aronszajn tree, Every Suslin tree is Aronszajn.
Definitions of a Suslin tree and Susline line
A proof that there is a Suslin tree if and only if there is a Suslin line
Knowledge of mutual implications between SH and other statements: diamond, V=L, GCH
Measurable cardinal
Definition of a measurable cardinal, two equivalent definitions of a kappa-complete ultrafilter
Proof that every measurable cardinal is strongly inaccessible.
Weakly compact cardinal
Proof of Ramsey theorem, definition of a weakly compact cardinal
Proof of Sierpinski's theorem: 2^kappa doesn't arrow (kappa^+)^2_2
Proof that every weakly compact cardinal is srongly inaccessible
Proof that every measurable cardinal is weakly compact
Tree characterisation of a weakly compact cardinal (and proof in one direction), understanding connection with Aronszajnness
Models of ZFC and absoluteness
Definition of absoluteness, proof that Sigma_0 formulas are absolue, Pi_1 formulas are downward absolute, Sigma_1 formulas are upward absolute
A list of examples of absolute formulas
Definition of definite terms and definite formulas, proof that definite formulas are absolute
Examples of non-absoluteness, draw the pictures how being a cardinal, being a power-set, being countable etc are non-absolute
A larger body of examples of absolute and non-absolute formulars.
Defining L
Definition of Sat
Definition of Def, proof that Def is a definite term
Definition of L_alpha and L
Proof that "x in L" is absolute
What holds in L
Proof of reflection principle, checking (axioms of ZF)_L
definition of a formula that well-orders L
Checking that ZF proves (V=L)_L
Proofs that Con(ZF) --> Con(ZFC) and Con(ZF) --> Con(ZF+ V=L)
ZF+ V=L imples GCH (will not be included in the exam)
Proof of Godel's Condensation Lemma
Proof that the size of H_{kappa^+} is 2^kappa
Proof that ZF+V=L implies that L_{kappa^+} = H_{kappa^+) (hence kappa^+= 2^kappa)
ZF + V=L implies Diamond (will not be included in the exam)
The lemma on elementary submodels of L_{omega_2}
Proof that ZF+V=L implies Diamond.
A bit more logic
Absoluteness of weak compactness (perhaps without proof)
V=L contradicts existence of a measurable (wihout proof)
Easy juggling with logical questions about consistency of certain theories
Linear orders
Counting linear orders of any cardinality, dense linear orders, discrete linear orders,
counting the number of discrete linear orders and dense linear orders of any cardinality, Cantor's theorem
(unique countable dense linear order),
the "long rational line" (counting the number of dense linear orders whose every initial segment is countable).
Exercise bunches.
Exam structure