Download Incompleteness Result

Incompleteness
General Description:
The positive story of the development of classical logic has culminated in Godel’s
completeness proof: “Whatever is true is provable” and this would encourage many
working mathematicians to continue their pursuit to prove or disprove those
historically well-known mathematical conjectures. However, the bad news is that
Godel also gives us the incompleteness proofs (1931), which in effect apply a
self-undermining “Godel sentence” which says that “I am not provable” (it’s fair to
draw an analogy between Godel’s sentence and instances of liar paradox) and thus
show mathematicians that for a sufficiently strong axiomatizable theory there are
some sentences, such as Godel sentence, which cannot be proved or disproved from
that theory. The impact made by the incompleteness results is so tremendous that on
the one hand, inspired by Godel, mathematicians have discovered later on quite a few
independent results—for instance, Canto’s continuum hypothesis is nowadays known
as being independent of ZFC, that is, it is neither provable nor disprovable from ZFC,
and the development is still at large; on the other hand, attempts have been made by
philosophers to try to give some insightful interpretations of the incompleteness
results—for example, it’s said that they show the limit of what machines can do.
This course is mainly designed to give a close study of the proofs of incompleteness
and then after having a good grasp of the technical stuff, we will take a look at some
philosophical issues triggered by incompleteness results. We will spend the first ten
meetings or so on articulating the technical matter and hopefully we can spare the rest
of our time for some philosophical issues.
Textbook:
Herbert B. Enderton, A Mathematical Introduction to Logic, New York: Academic
Press, 2001: Chapter Three.
Haim Gaifman, Notes on Godel’s Incompleteness Results (not available on the market
yet but the author has given his consent to our reproducing the notes for pure
academic use).
Suggested Readings:
Franzen, T., 2005, Gödel’s Theorem: An Incomplete Guide to its Use and Abuse,
Wellesley: A K Peters.
Haim Gaifman, “What Godel’s Incompleteness Result Does and Does Not Show”,
The Journal of Philosophy, 2000.
S. McCall, “Can a Turing Machine Know That the Godel Sentence Is True?”, The
Journal of Philosophy, 1999.
R. Penrose, Shadows of the Mind, Oxford, 1994.
Prerequisite:
Students should have at least some basic knowledge of set theory and model theory.
I suggest that those who want to take this course read Enderton’s 2.2, 2.6 and 2.7 first.
Schedule (tentative):
1. Introduction: A brief historical survey; Review of basics of first order logic.
2. The outline of Godel’s incompleteness proofs.
3. Number Theory; Some theories of sub-languages of number theory.
4. An axiomatized subtheory, AE of number theory; Representability of relations in
AE.
5. Basic rules of constructing relations and functions representable in AE;
Representability (in AE) and recursiveness.
6. A catalog of representable relations and functions; Formalizing the syntactics:
Godel numbers.
7. Representable sets of Godel numbers.
8. Fixed point theorem; Tarski’s undefinability theorem; Godel’s first incompleteness
theorem.
9. Midterm exam.
10. Strong undecidability of AE; Church’s Theorem; Godel’s second incompleteness
theorem.
11. Lob’s theorem; Rosser’s variant of the incomplete theorem.
12. *Incompleteness results and the mind of human beings (1)
13. *Incompleteness results and the mind of human beings (2)
14. *Incompleteness results and the mind of human beings (3)
15. *Incompleteness results and the mind of human beings (4)
16. Final exam.
* The prescheduled discussions of those meetings will be focused on some
philosophical issues from the suggested readings. But if students still have difficulty
understanding technical stuff, one or two of those meetings will be replaced by review
sessions.