The original question was about the Z' you mentioned before, i.e. Z without 6. > you can't just replace Z with this new set and reason about it using our mathe

The original question was about the Z' you mentioned before, i.e. Z without 6. > you can't just replace Z with this new set and reason about it using our mathematics Sure you can. Many things change and will break of course, and I agree counting doesn't work (which is the only reason I brought it up last time)—but that's exactly what the question is about: which specific things change and break. Some things would likely still continue to work though. For instance, you could construct something like I was describing above, which we might call Field-6, and do arithmetic and algebra with it. It wouldn't be as useful as a normal Field over the real numbers (for example), but that doesn't mean it wouldn't be a self-consistent thing describable within the framework of contemporary pure mathematics. You could also prove any number of theorems about it too—it's merely a question of why one would want to.