I am trying to understand bits and pieces of Lawvere's article [*Continuously Variable Sets; Algebraic Geometry = Geometric Logic*](https://www.dropbox.com/s/p8739l5bk01dpdc/Lawvere%20F.W%20-%20Continuously%20Variable%20Sets%3B%20Algebraic%20Geometry%20%3D%20Geometric%20Logic.pdf?dl=0). I'm not doing very well.

I know this is a lot to ask, but basically, I would like for someone to explain in more detail the whole construction described at the end of the paper. I was told the idea is to get a "free local ring", which is free simply because of the internal language of the corresponding sheaf topos which makes any ring locally free, but I really don't know what that means and I still don't understand the idea and details of the construction in the paper. I stress that I'm looking for a detailed explanation; I realize that the idea may be encapsulated in a short elegant statement, but I won't be able to decode it..

So... What the hell is going on there? :D