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Set theory prerequisites from 1998

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Math 771 Set Theory  Spring 2011  MWF 12:05

Text: Kenneth Kunen;  Set theory. An introduction to
independence proofs. North-Holland 1983, paperback,
ISBN-10: 0444868399

DESCRIPTION:  
This is a beginning course in set theory for
graduate students that already know some basic set theory:
ordinal and cardinal arithmetic, the Zermelo-Fraenkel axioms of
set theory, the Axiom of Choice, and Cantor's Continuum
Hypothesis.   Most students will have taken
Math 770 "Foundations of Mathematics" or the equivalent.

The Continuum Hypothesis is the statement that every subset of
real numbers is either countable or of the same cardinality
as the whole set of reals.  The Axiom of Choice is equivalent
to the statement that every set can be well-ordered.

Two highlights of the course are:

Theorem (Godel)  If the axioms of set theory are consistent,
then they remain consistent when we add the Axiom of Choice and
the Continuum Hypothesis.

Theorem (Cohen)  If the axioms of set theory are consistent,
then they they remain consistent if we add the negation of the
Axiom of Choice. It is also consistent with the axioms of set
theory to have the Axiom of Choice and the negation of the
Continuum Hypothesis.

These results are proved by constructing models of set
theory.  Godel's result is proved by taking an inner model
and Cohen's by creating a generic extension.

Topics will include:  Martin's Axiom, infinite combinatorics,
Godel's constructible sets, forcing and generic sets.