Mini-Course in Gödel's Incompleteness Theorems
For your fun, enrichment and intellectual stimulation, I am offering "live" online mini-courses in selected areas of
mathematics. The present feature is a mini-course in
Gödel's Incompleteness Theorems.
The Incompleteness Theorems, which were first published in the 1930s,
address the possibility of constructing an adequate formal axiom
system for arithmetic - i.e. a system of propositions that would
serve as an adequate formal definition of the set of positive
integers. The first Incompleteness Theorem shows that however one
chooses such an axiom system, it will always be possible to construct
a proposition which is a true statement about the positive integers,
but whose truth cannot be proved from the given axiom system. The
second Incompleteness Theorem shows that for any such axiom system for
arithmetic, if the system is consistent, its consistency cannot be proved within the
The issues with which these theorems are involved, and the questions
they raise, are of profound relevance to the philosophy of thought, mathematics, and science. They concern the nature of axiomatic
systems; the difference between logical coherence, logical
consistency, scientific consistency, and the intrinsic logic of
reality; the difference between operational adequacy of
scientific theories and their adequacy to Nature; the capability of Science to be more than
a model, or more than an axiomatic - to be an actual description of
reality; the nature of Mathematics and of Science.
The mini-course in Gödel's Incompleteness Theorems does not require any background in mathematics or
logic. They are taught live, using a conference-call
service and a chat program. Each course has 4 to 7 participants, and consists of 10 "live" weekly
sessions, each between 1.5 and 2 hours long. There is also a private Web forum
associated with each course. The total price per person, for the full course, is $200, if there are at least 4 participants. If the course is conducted one-on-one, the price is $500. There are no extra charges -
these prices already include all long-distance phone charges.
The rough "syllabus" is: 4 sessions to develop the methodological and
philosophical background, 4 sessions to discuss the 1st theorem, 1
session to discuss the 2nd theorem, and 1 session for summary and
general ramifications. The aim of the course is to present a clear, comprehensible
and philosophically uncluttered idea of the theorems and their role
in mathematics and philosophy.
If you are interested in taking part in this mini-course please let me know by
writing to the email address provided on top of this web-page . Tell me your name, the best way to contact you by email and by phone, and a bit about yourself (if you wish). And give me some indication as to what times are best for you. The typical schedule for the course is 9pm on a weekday. Would that be convenient for you? If not, what would be your preference?
Here is a very pleased email from a Gödel Mini-Course participant:
I'm writing to let you know how much I enjoyed your Godel Mini-course. You were a very effective communicator and your class notes were very illuminating. You brought a liveliness to what could have been a very dry subject - and you obviously know the material very thoroughly. I enjoyed your insights and examples, especially the one on Snork Theory being incomplete, it really brought the concept into focus.
I hope that you are successful in continuing to offer the class because it certainly has a niche to fill on the intellectual and mathematical landscape.
Good luck in all your endeavors and please, let me know if you decide to conduct a course on set theory ( or any other topic).
Thanks for a great class.