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Hmm, I think makes sense regarding counting things: it's nonsensical to use Z for counting things greater than five if you assert that it doesn't contain 6. Bu

Hmm, I think makes sense regarding counting things: it's nonsensical to use Z for counting things greater than five if you assert that it doesn't contain 6. But at the same time, that's just one use of Z. There are other uses of it besides counting. > Z exists independently of this universe or any universe. That's a pretty strong claim. Another possibility is that it's a purely human construction, and even in this universe it wouldn't exist if it weren't for us. (To be clear though, I am specifically referring to the set Z, which doesn't exist without the human creation of Set Theory. Maybe other universes would use something like Set Theory, but I think it would be hard to demonstrate one way or the other.)

opportune 9y ago
I talked about the use of counting to illustrate a point - not as "proof." What I think you're not understanding is that the set of integers without 6 is not Z. It is something else, and you can't just replace Z with this new set and reason about it using our mathematics. That's what I was trying to say when I said that Z exists independently of this universe or any universe: it is a "unique" (up to bijection) set generated by axioms that we could just as well define in a different universe or any universe, unless we are allowing universes that don't follow our laws of logic, in which case anything goes.
- westoncb 9y ago
  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.

opportune 9y ago

I talked about the use of counting to illustrate a point - not as "proof." What I think you're not understanding is that the set of integers without 6 is not Z. It is something else, and you can't just replace Z with this new set and reason about it using our mathematics. That's what I was trying to say when I said that Z exists independently of this universe or any universe: it is a "unique" (up to bijection) set generated by axioms that we could just as well define in a different universe or any universe, unless we are allowing universes that don't follow our laws of logic, in which case anything goes.

westoncb 9y ago
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.