The n-Category Café

Which Putnam exam is this from?

For those who don’t feel like bashing their heads over this, you can apply rot-13 to see a solution:

Abgvpr gung fhpu na nqqvgvir ubzbzbecuvfz zhfg or D-yvarne, juvpu vzcyvrf gung vgf tencu jvyy or n irpgbe fhofcnpr (bire gur svryq D) bs gur erny cynar. Rvgure vgf tencu vf pbagnvarq va n fgenvtug yvar (fb vf tvira ol zhygvcyvpngvba ol n erny fpnyne: gur E-yvarne pnfr), be ryfr vgf tencu unf gjb E-yvarneyl vaqrcraqrag ryrzragf k, l, jurapr pbagnvaf nyy engvbany yvarne pbzovangvbaf bs k, l, juvpu gura svyy bhg n qrafr fhofrg bs gur cynar.

It’s not as known as you might think. Andreas Blass proved that the assumption “every vector space over every field has a basis” + ZF implies the axiom of choice. A nice encapsulated proof may be found here, section 3.

But as to whether “every vector space over a fixed ground field $k$ has a basis” implies AC: as far as I know little is known. For example, it seems to be open in the case $k = \mathbb{Q}$. This MO thread discusses this point.

My gut tells me that there’s no way that a specific vector space like $\mathbb{R}$ having a $\mathbb{Q}$-basis would be nearly enough to prove AC. It’s just too restricted in scope. Also, while such a basis for $\mathbb{R}$ is somewhat nasty from the point of view of descriptive set theory (it can’t be Borel for instance), it need not be too nasty: see this MO answer by Joel David Hamkins for some information. Curiously, this type of thing came up in Tom’s other recent Café thread, where I mentioned this old result of Sierpinski that Hamel bases can’t be analytic sets (i.e., continuous images of Borel sets) – but they can be projections of complements of projections of closed sets, according to Joel’s answer!

I guess everyone here knows […]

I only know because I read Tom’s notes :-)

In one of life’s odd moments of synchronicity, I was just reading about this very topic earlier today, in Jeremy Gray’s The Hilbert Challenge (Oxford UP, 2001). I started sampling this volume on Google Books because one of my research topics made unexpected contact with Hilbert’s 12th Problem, and I have only the most casual acquaintance with that whole area.

I’m glad you find it appealing!

I advertised it not only to staff and PhD students, but also to our 4th- and 5th-year undergraduates, telling them that they should be well-placed to understand it if they had a good grasp of our compulsory 3rd-year course Honours Analysis. Only a small portion of that course is actually needed for what we’re doing with functional equations, but I thought that was about the right level in terms of general mathematical experience.

There will be bits and pieces of more advanced stuff that we’ll need soon, but again it’s the kind of thing that most of our mathematics undergraduates pick up in their 3rd or 4th year. Of course, education systems vary enormously from country to country and even university to university, but I hope that gives a rough indication.