John Baez wrote: > In article <email@example.com>, > <firstname.lastname@example.org> wrote: > >>Another way to understand the analogy between sets and vector spaces >>is via the language of matroids. Sets correspond to "free matroids" >>and vector spaces correspond to "representable matroids." How much >>of what you say about the one-element field can be understood in >>terms of matroids? > > I don't know, since I don't know much about matroids. But, from > what little I know, a lot of projective geometry fits nicely into > the matroid framework. That's promising, since the stuff I'm > talking about is closely related to projective geometry. > > According to what you say, both for free matroids and "representable > matroids" (whatever those are) over a given field we get one of these, > up to isomorphism, for each natural number n - the cardinality or > dimension. Are there other nice series of matroids, one for each > natural number? That might lead to some fun stuff.
The following reference might contain an explanation of your observation:
B. Zilber, Uncountably categorical theories. Transl. from the Russian by D. Louvish. Transl. ed. by Simeon Ivanov. (English) Translations of Mathematical Monographs. 117. Providence, RI: American Mathematical Society. vi, 122 p. (1993). ISBN 0-8218-4586-1/hbk
One of the main results is the following:
Trichotomy Theorem: Every uncountably categorical theory is of one of the following types, according to the geometries it realises: (1) field-like (2) module-like (3) disintegrated