The problem is that for whatever reason Microsoft decided that there should be three distinct types of properties: read-only, write-only, and read-write, only one of which may exist with a given signature in a given context; properties may only be overridden by identically-declared properties. To do what you want it would be necessary to create two properties with the same name and signature--one of which was read-only, and one of which was read-write.
Personally, I wish that the whole concept of "properties" could be abolished, except that property-ish syntax could be used as syntactic sugar to call "get" and "set" methods. This would not only facilitate the 'add set' option, but would also allow for 'get' to return a different type from 'set'. While such an ability wouldn't be used terribly often, it could sometimes be useful to have a 'get' method return a wrapper object while the 'set' could accept either a wrapper or actual data.
