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If there were no number 6, there could be no infinite countable sets, because all infinite countable sets have a bijection to Z, and 6 is in Z. Furthermore ther

If there were no number 6, there could be no infinite countable sets, because all infinite countable sets have a bijection to Z, and 6 is in Z. Furthermore there could be no finite sets with cardinality greater than 6 because those have a bijection to Z_n for some n in Z. That would be true in both a mathematical sense and a physical sense, since you could still physically represent the quantity 6 with 6 particles. So any "world" with no 6 could have at most 3 bits of information in it. It's really a nonsensical question because 6 is both an abstract concept - hence the reasoning about infinite and finite sets - and a real quantity you can have of something. The normal arithmetic operators wouldn't even be well-defined. The only way to make this universe consistent is if you made an exception such as 5 + 1 = 7 and 7 - 1 = 5, but in that case you have simply replaced the symbol of the number k with the symbol of the number k-1 for all k > 6. It's just an absurdity, and in my opinion a poor question to write about for a non-mathematical audience, because it will make people confused. It's like asking the question what if there were no sets, or what if there weren't addition. Math is truth not limited by the constraints of our physical universe; it is an absolute. There has to be a concept of 6 for there to be mathematics - or anything - at all.

westoncb 9y ago
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

westoncb 9y ago

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