Re: defining sets from abelian groups

- I want to say something along the lines of "You can define abelian - groups in the internal logic of Set, but you can't define sets in the - internal logic of Ab." - - Is this intuition supportable? Ideally there would be a precise - technical sense in which it is provably the case. - - Vaughan Pratt - - I don't know, since I don't understand these issues sufficiently well, but if you can define the free abelian group cotriple, you can then define the coalgebra category for that cotriple and that is the category of sets. In fact, that is a ! operation, in Girard's sense, on the fragment of linear logic that Ab satisfies. (This last was an observation of Jim Otto's.) Michael Barr