Like most people in mathematics, you are probably a devotee of astrology. If you are also a Scorpio, you’re in luck, since this is your horoscope according to the Onion:

Love means different things to different people, but you’re the only one for whom it means that to every w-consistent class K of formula there correspond recursive class-sign r (on free var. v), such that neither (v Gen r) nor ~(v Gen r) belong to Flg (K).

Via Computational Complexity.

It gets worse. A countably infinite tree can fail to have an infinite branch if you allow a node to have infinitely many successor nodes: the tree becomes infinitely wide rather than infinitely tall. This suggests that if we impose a growth condition that prevents an uncountable tree from becoming uncountably wide, maybe we can force it to have an uncountable branch.

Requiring that each node have at most countable successor nodes is not sufficient. Aronszajn trees only split a countable number of times at each node, but still has no uncountable branch. A stronger condition is that the tree has no uncountable antichain. An antichain is a subset of the tree such that no node in the set is the descendent of any other node. The size of the largest antichain gives a rough measure of the width of the tree. If we force the tree to have a countable width by this measure, then possibly this will force the tree to have an uncountable branch. This is a much stronger property than requiring each node to have only a countable number of successors. For example, an countable binary tree has a countable antichain, but it has lots of countable branches.

Does it work? Intuition from the countably infinite frequently fails when we consider uncountable sets, as the case of Aronszajn trees shows. At the same time, we have imposed a very strong condition, so perhaps that would be enough to settle it. So the result is plausibly true, and plausibly false. Which is it?

Neither. The existence of an uncountable tree with neither an uncountable antichain or an uncountable branch, known as a Suslin tree, is undecidable by the axioms of set theory. The question is equivalent to the Souslin problem, which has been shown to be independent of ZFC. (The axiom of constructability implies the existence of a Suslin tree. Assuming Martin’s Axiom and that the continuum hypothesis is false implies the non-existence of a Suslin tree. Both setups are known to be consistent with ZFC, which means that the existence of a Suslin tree must be independent.)

Uncountable trees are bad news.

A well-known theorem about infinite trees that if a tree only grows a finite amount at each node, it must have an infinite branch. This is known as König’s lemma. Obviously, an uncountable tree that only grows a countable amount at each node must have an uncountable branch, right? Amazingly, the answer is no. Counterexamples, now known as Aronszajn trees, were constructed by Nachman Aronszajn. Keith Devlin has more.

Andrey Burkov asked me to let the world know about his website Texify. He’s running mimeTeX, so that you don’t have to. You can use the site to generate images from TeX code for your site. You can link to the site rather than hosting the image yourself, and the site even supports directly encoding TeX into the URL of the image. For example, this image

was generated on the fly by his site from TeX I embedded in the URL.

MimeTeX is a tool that converts TeX formulas into GIFs. We posted about it last year.

Peter Woit has more on the resignation of the K Theory editorial board.

I once had an independent study reading Ayoub’s book on analytic number theory. In that book, I remember the Prime Number Theorem being a hard slog. It turns out that D. J. Newman published a short 6-page proof in 1980. The proof requires complex analysis at the level of an undergraduate course.

This message by Joe Shipman to the Foundations of Mathematics mailing list contains a link to a write-up of the proof by Zagier.

Via Antimeta.

A group of rabble-rousers and anarchists have started EUREKA Science Journal Watch, a wiki-based site to collect information about journal publishing practices. Their aim is to replace the current academic publishing system with one based on open access. I can only assume that once they have finished destroying science as we know it, that they will move on to destroying all civilization, and then life on Earth.

Alexandre Borovik has posted an update to the unfolding Education without Permission story. The school has been reopened, but the Turkish authorities still plan on pressing charges for “education without permission” against the school’s founder, Ali Nesin. Nesin plans on defending himself on constitutional grounds. (Interestingly, the Turkish constitution explicitly grants the right to teach and learn arts and sciences on constitutional grounds.)

I think the worst terminology in all of mathematics may be that of G-delta and F-sigma sets. Even though I learned the meaning of them years ago, they still convey no information to me at first glance. I just looked at the definitions of perfectly normal space and Baire set, and without concentrating on that mysterious G-delta they mean nothing to me. Good terminology should provide a hint as why the definition isn’t some other way. Why not F-sigma sets? G-delta-sigma sets? G-sigma-delta sets?

Which is too bad, because in both instances the property is completely natural. The perfectly normal spaces are exactly those spaces where every closed set is a zero set of a continuous function. The Baire sets are for doing measure theory when you’re outside the friendly confines of a perfectly normal space. To integrate real-valued bounded continuous functions, we only need to consider Baire sets. Other sets are uninteresting from the point of view of measure theory. But in both cases it’s easier to work back from the desired property to the definition than vice versa.