Aims/Objectives. This course seeks to introduce the student to the (then) revolutionary ideas of Georg Cantor (the first mathematician to make sense of 'the infinite'), but not by the all-too-easy, but fundamentally shallow 'definition-theorem-proof' textbook route. Rather the course attempts to explore his ideas, initially by posing certain questions relating to 'Hilbert's Hotel' - questions which may be posed even to those having no mathematical vocabulary whatsoever. (It would be my hope that students would themselves pose such questions to their own non-mathematical friends, family, ... , as a means of showing that one doesn't necessarily need to be mathematically literate to consider some non-trivial mathematical questions...)
Learning Outcomes. At the end of this course a student should be familiar with, not only the central achievement of Georg Cantor, but have acquired working facility - with hand performed and maple assisted computations - with a number of related technical mathematical topics: reducible and irreducible polynomials (over Q and R), algebraic and transcendental numbers, decimal and other base expansions, continued fraction expansions of real numbers (and in particular those of real quadratic irrationalities).
I consider Georg Cantor to be one of the greatest mathematicians, whose work may be read or studied at many levels.
The Real Numbers and Cantorian Set Theory is a new course that I introduced in the academic year 2002-2003. I didn't lecture the course in the standard way; rather I sat with my students, in my office, and asked them to consider certain questions... The informed reader will suspect what I was trying to do...
At least I will tell you how we began: I asked them to imagine an hotel, an 'infinite' hotel (of course, eventually, the word 'countable' was used for the number of rooms...), with one person in every room. Lights out, everyone in bed... (That hotel, is, of course, 'Hilbert's Hotel, though it may also be Gamow's hotel? My own introduction to this playful way of encountering cantorian ideas was in Vilenkin's great book, Stories about Sets.)
Then, in the middle of the night, one stray person turned up, wanting a room. The hotel only allows at most one person to occupy a room... Question: can the 'extra' person be accommodated without asking someone to leave the hotel?...
Well, as you may imagine, we had some fun...
Only bit by bit did I introduce any 'mathematics'... , the
tidied up version of most - but not all - of which may be read in the following notes (some 29 A4 pages):
Some related notes:
polynomials (Word, zip, 53KB)
Other related Maple notes (on decimal expansions of real numbers)
expansions (active Maple mws worksheet, 37KB)
I also did some Maple work with them on continued fraction expansions, but I didn't save any of it. Next year...
Finally, yesterday my students had their 3-hour end-of-year exam, which I can now put into the public domain:
Exam paper (Word, zip, 18KB)
After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com