Set theory (Math 320)
Spring semester, 2006/7
Added, December 15, 2009: I wrote up some corrections to the notes distributed for this course:
I prepared a new edition of the notes for 2008/9.
Added, May 12, 2007: Some notes on ordinal numbers are available.
We meet Monday, 13.40–15.30, and Thursday, 10.40–11.30, in M105. Attendence is expected. The first exam is in class, March 26 (Monday). The second exam is May 7.
I recall here the catalogue-description of the course:
Language and axioms of set theory. Ordered pairs, relations and functions. Order relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Cardinality and arithmatic of cardinal numbers. Axiom of choice, generalized continuum hypothesis.
Set-theory is useful to mathematics in the following way. We establish certain properties of sets, usually by means of axioms, such as the so-called Zermelo–Fraenkel system of axioms. Then we can define various standard mathematical structures (such as the ordered field R of real numbers); we can prove the various properties of these structures, without having to introduce new assumptions. In this way, set-theory provides a uniform foundation for mathematics.
Set-theory is also worth pursuing for its own sake. The development of set-theory involves the creation of certain new structures (such as the class ON of ordinal numbers, or the class L of constructible sets) that are interesting in themselves. Set-theory gives us certain useful techniques (such as proof by induction and definition by recursion).
Number-theory contains theorems (such as the so-called Fermat's Last Theorem) that are easy to understand, but almost impossibly difficult to prove. Some theorems of set-theory are harder to understand (and also difficult to prove), but are mind-blowing in their connotations. For example, R is strictly greater in cardinality than the set ω of natural numbers; in a word, R is uncountable; however, the Zermelo–Fraenkel axioms do not determine whether R has the least uncountable cardinality.
I prepared notes for the course and printed them for distribution.
The notes are entirely rewritten from the notes used in 2003.Further references include:
- Yiannis Moschovakis, Notes on Set Theory (second edition), (Springer, 2006) (available for download at METU)
- Azriel Levy, Basic Set Theory (Dover, 2002; original publication: Springer, 1979)
- (many other books)
