An introduction to independence proofs
This is a first graduate course in set theory. The primary topic is Hilbert's First Problem - the independence of the continuum hypothesis (CH).
In the first half of the 20th century it was shown that most of mathematics can be formalized within set theory. The standard axiomatic system is Zermelo-Fraenkel with choice (ZFC). Then Gödel's Incompleteness theorem showed that mathematical truth cannot be settled by ZFC. The most famous example is the continuum hypothesis (CH), which states that every infinite set of reals is either countable or equinumerous with the set of all real numbers. By results of Gödel and Paul Cohen (who invented the breakthrough method of forcing) CH was then shown to be independent of ZFC.
We will cover the axioms of set theory (ZFC), ordinals, cardinals, Gödel's Constructible Universe L, and his proof that ZFC+CH holds in L. That implies that CH cannot be refuted from ZFC. Then we will go over Cohen's method of forcing and use it to show the consistency of the negation of CH with ZFC i.e. that one cannot prove CH from these axioms.
Optional: Kenneth Kunen, Set Theory, College Publications, 2011.
graduate level mathematical maturity and some set theory