On Wed, Jul 13, 2011 at 2:16 AM, Marta Bunge <martabunge@hotmail.com> wrote:
Bob Pare and I wanted to posit something like the existence of a "representative (or universal) cover" as an axiom (RC), such that GT => RC => ASC, and also (ET + AC) => RC. We hoped to show that RC had good properties, and more importantly, that Eff satisfied it but this did not work.
That's a very interesting question! There seem to be a lot of axioms of this flavor, which say in various different ways that AC fails "in only a small way". Some that I am aware of include: * Small Violations of Choice (Blass): http://nlab.mathforge.org/nlab/show/small+violations+of+choice * Small Cardinality Selection (Makkai): http://nlab.mathforge.org/nlab/show/small+cardinality+selection+axiom * Axiom of Multiple Choice (Moerdijk & Palmgren): http://nlab.mathforge.org/nlab/show/axiom+of+multiple+choice * Weakly Initial Sets of Covers (Roberts): http://nlab.mathforge.org/nlab/show/WISC * The ex/lex completion of Set (or the topos in question) is well-powered * The ex/lex completion of Set is a topos, i.e. Set has a generic proof Some of these hold in any Grothendieck topos, but others apparently need not, and I have no idea which of them might hold in Eff. It would be interesting to know if any of them imply ASC (or André's proposed strengthening thereof). Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]