Reverse Mathematics of Matroids
Abstract
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for $\Sigma^0_2$ formulas.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1604.04912
 Bibcode:
 2016arXiv160404912H
 Keywords:

 Mathematics  Logic;
 03B30 (Primary) 03F35;
 05B35 (Secondary)
 EPrint:
 20 pages