I mentioned that Glanzberg’s paper focuses on Williamson’s version of Russell’s paradox for interpretations. I can’t say that I find that version very illuminating, but there it is. But it does shape Glanzberg’s discussion, and he tells the story about background domain expansion in terms of someone’s reflecting directly about the interpretation of their own language. But I don’t think that this is of the essence, nor the clearest way to present a discussion about what Dummett would call indefinite extensibility.
What is central to the discussion is Glanzberg’s reflections on how far we should iterate the expansion of our domain of “absolutely everything”, once we grasp the (supposed) Dummettian imperative to start on the process. Dummett’s talk about indefinite extensibility suggests that he thinks that there is no determinate limit point (which, I take it, isn’t to say that the expansion definitely goes on for ever, but that there is no point where we have a clear reason to stop). Now, Glanzberg by constrast, things here might be reason just to embrace iteration up to the first non-recursive ordinal, or up to the first α + 1 ordinal, or up to the first α+ ordinal. He then says
In considering multiple options, I do not want to suggest that there is nothing to distinguish among them … Rather, I think the moral to be drawn is that we do not yet know enough to be certain just how far iteration really does go.
But, once we play the Dummettian game, I just don’t see why we should think that there will be a determinate answer here, and certainly Glanzberg gives us no clear reason to suppose otherwise.