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I think your proof using the bijection amounts to: it wouldn't work because 6 wouldn't be in Z—but that is the essence of the proposition here: "what if 6 weren

I think your proof using the bijection amounts to: it wouldn't work because 6 wouldn't be in Z—but that is the essence of the proposition here: "what if 6 weren't in Z?" So you end up just dismissing the question. Sure, it's a fairly arbitrary idea that isn't really going to go anywhere, but maybe it can at list be a bit more interesting. For example, what if we treated operations resulting in what would be the 7th member (counting from zero) of Z (i.e. '6') the same we treat division by zero? If we are constructing the system axiomatically, we can follow the the pattern with Fields, looking at their definition of multiplicative inverses (https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_definition https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_de...), which effectively excludes the possibility of division by zero. There's probably a similar way of patching axioms related to addition and multiplication so as to preclude the possibility of arriving at '6' through application of those operations.

opportune 9y ago
Ultimately the problem with reasoning like this is that it's so nonsensical that things get confusing. For example, I'm sure we can agree that 6 is in Z. If we remove 6 from Z, we are talking about another set: it's no longer Z but Z', and because 6 is in Z but not Z' Z' != Z. Sure we can define new operations and such on this set Z' but there is still only one Z. That's different from asking about a world or a universe without 6 though. Z exists independently of this universe or any universe. In a physical universe with discrete elements, you'll probably want to count those discrete things, and for that you have to use Z. Basically the argument I was making before was that you're no longer counting things if you don't have 6, because you count things with N, and 6 is in N. If you count things in N' which doesn't have six, whatever number you use to count things after 5 is 6 or you're not counting. The main thing I'm saying is that you can make define a set that doesn't contain 6 and for which operations are defined differently, but then it's not Z, and there has to be a Z. Also, note that we can derive/define the integers (and essentially all of what we call mathematics) from these axioms: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory. If theres no number 6, we need to pick a different set of axioms. At that point you're arguably not even doing mathematics because these axioms are so ubiquitously general across all of math.
- westoncb 9y ago
  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

Ultimately the problem with reasoning like this is that it's so nonsensical that things get confusing. For example, I'm sure we can agree that 6 is in Z. If we remove 6 from Z, we are talking about another set: it's no longer Z but Z', and because 6 is in Z but not Z' Z' != Z. Sure we can define new operations and such on this set Z' but there is still only one Z. That's different from asking about a world or a universe without 6 though. Z exists independently of this universe or any universe. In a physical universe with discrete elements, you'll probably want to count those discrete things, and for that you have to use Z. Basically the argument I was making before was that you're no longer counting things if you don't have 6, because you count things with N, and 6 is in N. If you count things in N' which doesn't have six, whatever number you use to count things after 5 is 6 or you're not counting. The main thing I'm saying is that you can make define a set that doesn't contain 6 and for which operations are defined differently, but then it's not Z, and there has to be a Z. Also, note that we can derive/define the integers (and essentially all of what we call mathematics) from these axioms: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory. If theres no number 6, we need to pick a different set of axioms. At that point you're arguably not even doing mathematics because these axioms are so ubiquitously general across all of math.

westoncb 9y ago
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.