Timeline for Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?
Current License: CC BY-SA 2.5
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 26, 2011 at 16:24 | comment | added | Asaf Karagila♦ | @Konrad: Wow, I completely forgot about these two comments. It's been a while since I played with this idea, but recently a mistake was found in a related claim, so I am checking the details again, if there is no problem I will add this as an answer (it is not long at all) | |
| Oct 26, 2011 at 14:39 | comment | added | Konrad Swanepoel | @Asaf: Interesting! Do you have a preprint? | |
| Jun 6, 2011 at 19:48 | comment | added | Asaf Karagila♦ | @Konrad: Of course it carries over! It is very simple, a simple generalization of Blass construction (generalizing Lauchli) still yields a vector spaces without non-scalar automorphisms, and from here the proof by Todd carries over completely. | |
| Jun 6, 2011 at 19:45 | comment | added | Asaf Karagila♦ | @Konrad: Something that I am working on nowadays shows that in Lauchli models it is consistent to have $DC_\kappa$ for arbitrarily high $\kappa$. I'm not sure that Todd's proof will carry though. | |
| Dec 15, 2010 at 12:21 | comment | added | François G. Dorais | The permutation model used by Andreas is in fact due to H. Läuchli. [Auswahlaxiom in der Algebra, Comment. Math. Helv. 37 1962/1963] This is the standard model for the existence of a vector space without a basis. (Note that there is nothing special about $\mathbb{F}_2$ or rather $\mathbb{F}_4$ in the construction.) The vector space in question has the curious property that all of its proper subspaces are finite dimensional; this gives a very quick proof of the desired result. | |
| Dec 15, 2010 at 12:06 | comment | added | Todd Trimble | @Konrad: unfortunately, I do not. I was really hoping that Andreas would see your questions (which I think are great), since he really is a resident guru on this sort of question. | |
| Dec 15, 2010 at 11:42 | comment | added | Konrad Swanepoel | Very nice. Do you know how badly AC fails in Blass's model? | |
| Dec 14, 2010 at 16:51 | history | answered | Todd Trimble | CC BY-SA 2.5 |