The Real Numbers and Cantorian Set Theory

(joint BA course #1)

**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):

Real
Numbers and Cantorian Set Theory (pdf format 736 KB)

Some related notes:

Irreducible
polynomials (Word, zip, 53KB)

Irreducible
polynomials (Wordrtf, zip, 81KB)

Other related **Maple** notes (on decimal expansions of real
numbers)

Decimal
expansions (active Maple mws worksheet, 37KB)

Decimal
expansions (html text version of Maple worksheet, with 'images' folder, as
all Maple outputs are converted to gif files)

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:

2002-2003
Exam paper (Word, zip, 18KB)

2002-2003
Exam paper (Wordrtf, zip, 16KB)