Filter quantifiers

I have been supervising an undergraduate student in an independent study in topology this semester. We have just finished the Stone–Čech compactification, and the semester is ending, so I want to end with an ultrafilter based proof of Hindman’s theorem. There are several good proofs of this sort online, including Hindman’s Theorem Via Ultrafilters (PDF) by Leo Goldmakher and Ramsey Theory (PDF) by Imre Leader (lecture notes transcribed by Stiofáin Fordham). The latter is particularly nice because it uses the formalism of filter quantifiers.

Continue reading

Talk on Reverse Mathematics and the Modal Logic of Reverse Mathematics

This is a transcription of notes from a talk I gave on November 1, 2013 to the interdisciplinary logic seminar at the University of Connecticut. I gave a general introduction to Reverse Mathematics and then spoke about my work with Alaeddine Saadaoui and Sean Sovine on the logic of implications between subsystems in in Reverse Math. This post is essentially my speaking notes for the talk.
Continue reading

Internal combinatorics and uniform reducibility

This post is a set of notes from a talk I gave on December 5th for the discrete mathematics seminar at Marshall University. I want to argue that logical analysis can reveal the “internal combinatorics” of theorems, using some recent results of Dorais, Dzafarov, Hirst, Mileti, and Shafer [DDHMS 2012] as a particular example of the process.
Continue reading

Quiz on public peer review

I have been required to complete a “responsible conduct of research” training module by the research office at my school. The reason I am commenting is that I was asked to answer the following question “true” or “false”. This is not meant to be an opinion, it is from a quiz on which I am supposed to maximize my score.

True or false: “A good alternative to the current peer review process would be web logs (BLOGS) where papers would be posted and reviewed by those who have an interest in the work.”

I thought that would be interesting to people who follow Boole’s rings. The correct answer, of course, was “false”. Here is the rationale they gave:

“Although the peer review process is evolving, the described system would probably not work very well. It is likely that the peer review process will evolve to minimize bias and conflicts of interest. It is, in the best interest of everyone involved in the research enterprise that the scientific review process be fair and rigorous.”

Just throwing it out there….

Computable roots of computable functions

Here are several interesting results from computable analysis:

Theorem 1. If $f$ is a computable function from $\mathbb{R}$ to $\mathbb{R}$ and $\alpha$ is an isolated root of $f$, then $\alpha$ is computable.

Corollary 2. If $p(x)$ is a polynomial over $\mathbb{R}$ with computable coefficients, then every root of $p(x)$ is computable.

Theorem 3. There is a effective closed subset of $\mathbb{R}$ which is nonempty (in fact, uncountable) but which has no computable point.

Theorem 4. There is a computable function from $\mathbb{R}$ to $\mathbb{R}$ which has uncountably many roots but no computable roots.

Continue reading

An incompleteness theorem for βn models

My first paper was “An incompleteness theorem for $\beta_n$ models” with Stephen Simpson [1]. It’s a short paper, but the idea is very pretty. We know that the incompleteness theorem implies there are strange models of arithmetic, but these models often seem mysterious, and it’s hard to see what useful properties they can have. But now suppose that a theory of the form $A+B$ meets the hypotheses of the incompleteness theorem, and moreover this theory proves its own consistency, so that $A+B$ is inconsistent. It follows that if $A$ is true (that is, true in the standard model) then $B$ must be false. In this way, we can use the incompleteness theorem to prove facts about the standard model rather than about nonstandard ones. The idea is originally due to Harvey Friedman in his thesis, I believe.
Continue reading

The logic of Reverse Mathematics

This post is about a research idea I have been thinking about which is quite different from my usual research. It’s an example of a project with an “old fashioned” feel to it, as if it could have been studied 50 years ago. It’s almost a toy problem, so I haven’t spent too long digging through references yet. For all I know it was solved 50 years ago. Continue reading