The statement that the cardinality of the real numbers is the next infinite cardinality after the cardinality of the natural numbers, namely Cantor's Continuum Hypothesis (CH), was at the top of David Hilbert's 1900 list of the most significant open problems in mathematics. The work of Kurt Gödel (1940) and Paul Cohen (1963) demonstrate that CH is neither provable nor refutable from the standard axioms of set theory (ZFC). This independence result has significant implications in mathematics, logic, and philosophy. Following a nutshell overview of background material in logic and set theory (including Gödel's Completeness and Incompleteness Theorems, Zermelo-Fraenkel axioms, Axiom of Choice, ordinal and cardinal number systems), this course explores independence results in set theory in general, as well as some of the key methods for proving them. Topics include Gödel's model L for ZF with CH, and Cohen's method of forcing for a model of ZF with the negation of CH. Time permitting, additional topics may include independence results associated with large cardinal axioms, existence of measurable sets, and axioms of determinacy. 2 unit option is only for Philosophy PhD students beyond the second year.
2-4 units · Letter or Credit/No Credit
The statement that the cardinality of the real numbers is the next infinite cardinality after the cardinality of the natural numbers, namely Cantor's Continuum Hypothesis (CH), was at the top of David Hilbert's 1900 list of the most significant open problems in mathematics. The work of Kurt Gödel (1940) and Paul Cohen (1963) demonstrate that CH is neither provable nor refutable from the standard axioms of set theory (ZFC). This independence result has significant implications in mathematics, logic, and philosophy. Following a nutshell overview of background material in logic and set theory (including Gödel's Completeness and Incompleteness Theorems, Zermelo-Fraenkel axioms, Axiom of Choice, ordinal and cardinal number systems), this course explores independence results in set theory in general, as well as some of the key methods for proving them. Topics include Gödel's model L for ZF with CH, and Cohen's method of forcing for a model of ZF with the negation of CH. Time permitting, additional topics may include independence results associated with large cardinal axioms, existence of measurable sets, and axioms of determinacy. 2 unit option is only for Philosophy PhD students beyond the second year.