Math 116 provides an introduction to the basic concepts and results of mathematical logic and set theory. Math 116B will be devoted to computability theory and the incompleteness theorems.

(Additional topics may include the behavior of countable models and Hilbert’s 10th problem.)

Our approach to incompleteness will be somewhat non-standard and will allow us to discuss subsystems of second-order arithmetic.

Grading Policy: The grade for this course will be based on homework assignments. There will be no exams.

Solutions to homework problems should be written individually, although collaboration is allowed. All references used to solve a problem should be explicitly mentioned, including those students you collaborated with. You cannot look up solutions from any source.

No late submissions of solutions are allowed, except for medical problems (note needed from the health center) or serious personal difficulties (note needed from the Dean’s office).

Please try to solve as many problems as it seems reasonable from each set. Let me know if you find some problems to be too hard or too easy or to contain mistakes. Feedback is greatly appreciated.

Textbook: There is no required textbook. The following suggested references may be useful:

R. Cori and D. Lascar. Mathematical logic. A course with Exercises (Part II), OUP, 2001, ISBN 0198500505

T. Franzén. Inexhaustibility, ASL (A K Peters), ISBN 0198500505 J. Shoenfield. Mathematical logic, ASL (A K Peters), ISBN 1568811352S. Simpson. Subsystems of second order arithmetic, Springer, ISBN 3540648828

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta