Fall 2026
Dima Sinapova
Subtitle:
An introduction to independence proofs
Course description:
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.
Text:
Optional: Kenneth Kunen, Set Theory, College Publications, 2011.
Prerequisites:
501 and a course in point set topology like 411 or 502.
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Fall 2024
Dima Sinapova
Subtitle:
An introduction to independence proofs
Course description:
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.
Text:
N/A
Prerequisites:
This class should be accessible to any graduate student.
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Spring 2023
Dima Sinapova
Subtitle:
An introduction to independence proofs
Course description:
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.
Text:
None
Prerequisites:
This class should be accessible to any graduate student.
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Spring 2020
Grigor Sargsyan
Subtitle:
Proofs of determinacy
Course description:
Thinking of reals as point in the Baire space N^N, consider the two player game with payoff set A subset N^N in which players collaborate to produce a real x and player I wins it x is in A.
Axiom of Determinacy is the statement that all games as above are determined, i.e., one of the players has a winning strategy.
AC implies that AD is false, but definable versions of AD are true. For example a classic theorem of Martin says that all Borel games are determined.
In this course we will develop techniques for proving the determinacy of definable games. Along the way we will develop many tools needed for doing research in set theory.
Text:
None
Prerequisites:
graduate level mathematical maturity and some set theory
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Spring 2019 Simon Thomas
Subtitle:
Set-theoretic forcing: an introduction to independence proofs
Course description:
This is an introductory course on proving independence results in set theory. Here a statement S is said to be independent of set theory if S can neither be proved nor disproved from the classical ZFC axioms of set theory. For example, we will show that the Continuum Hypothesis CH is independent of set theory.
Text:
Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam
Prerequisites:
A knowledge of basic set theory, including cardinals, ordinals and the axiom of choice.
Schedule of Sections
Previous Semesters:
- Spring 2020 Prof. Sargsyan
- Spring 2019 Prof. Thomas