These are lecture notes from an introduction to axiomatic set
theory taught
by
Scott
Weinstein in the spring of 2009, typed up
by
me
and
Vilhelm
Sjoberg. I make no guarantees whatsoever as to their
completeness or correctness!
A PDF for each individual lecture is provided below; you can also download
all the lecture notes
in
one big PDF.
| 1/21 |
Well orderings |
| 1/26 |
Ordinals,
transfinite induction |
| 1/28 |
Transfinite
recursion, cardinals |
| 2/2 |
Cardinals (Axiom of Choice,
cardinality, normal functions, cofinality, cardinal
arithmetic) |
| 2/4 |
More cardinals (regularity, the
Continuum Hypothesis, Koenig's theorem) |
| 2/9 |
The real line (partial isomorphisms,
Cantor's back-and-forth theorem, uniqueness of the reals, the
Cantor set) |
| 2/11 |
The real line, part II
(perfect sets, Cantor-Bendixson theorem) |
| 2/16 |
Introduction to relative
consistency results (Axiom of Regularity, transitive closure,
relativization, absoluteness) |
| 2/18 |
A digression on
absoluteness (elementary substructures, Mostowski's Collapsing
Lemma, nonstandard (non-well-founded) models, completeness and
compactness of first-order logic, Lowenheim-Skolem Theorem,
Skolem's paradox) |
| 2/23 |
Proof of the Lowenheim-Skolem
Theorem and related results |
| 2/25 |
More relative
consistency: consistency of ZF - Reg implies consistency of
ZF. |
| 3/2 |
Strongly inaccessible cardinals and ZF. |
| 3/16 |
Midterm exam review. |
| 3/18 |
The reflection principle. |
| 3/23 |
The constructible
hierarchy, formal definition (encoding formulas as sets,
satisfiability, Delta-1) |
| 3/25 |
The constructible
hierarchy, part II (More Delta-1, absoluteness of L, almost
universal, L models ZF) |
| 3/30 |
The constructible hierarchy, part III (inner models, ZF + (V=L)
proves AC) |
| 4/1 |
The constructible
hierarchy, part IV (ZF + (V=L) proves GCH) |
| 4/6 |
Independence of CH from ZFC
(partial orders, filters, P-genericity) |
| 4/8 |
Independence of CH (cardinal
preservation, antichains, ccc, quasi-disjointness) |
| 4/13 |
Independence of CH (more ccc
and preservation of cardinals, P-names) |
| 4/15 |
Independence of CH (generic
extensions, properties of generic extensions, forcing, Truth, Definability) |
| 4/20 |
Independence of CH (M[G] is a
ctm, digression on Kripke-Platek set theory, axiom of Collection,
and computability) |
| 4/22 |
Independence of CH (finish
proof that M[G] is a ctm), Ramsey's theorem |
| 4/27 |
Ramsey cardinals |