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Comment by OJFord | Hacker News Reader

> If there’s no 6, said Caroline Turnage-Butterbaugh, a math professor at Duke University, then there can’t be a 7, 8 or 9 — or, really, any number greater than 5. Say you have a pile of seven items. You could take one away, ending up with the-number-formerly-known-as-6.

This seems like a weak argument: say you have a pile of five items; you could add one, and then we have six after all.

I don't see why subtraction should be treated any differently?

I thought this was going to be more interesting than 'imagine the world in base 6' - what're the ramifications of a 'hole' in the middle? I recall enough to know that fields allow for it, but are the consequences at all interesting?

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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 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.

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_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.

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. 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.

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.)

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.

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.

This seems like a weak argument: say you have a pile of five items; you could add one, and then we have six after all.

You presume the existence of a sixth item, so of course you can construct it.

There are multiple interpretations to the question, clearly.

Removing a tick mark from an infinite number line would just cause all the other tick marks to slide closer together like billiard balls, no?

Or is the question really about what happens if we render six as a forbidden, unaddressable element? Maybe equations normally resulting in an integral six, or involving an integral six return an error stating: undefined? Not unlike a divide by zero error?

Consider rationalizing division by zero. One could interpret the outcome of such an operation as always resulting in zero, instead of an error, but what would that mean when modeling real world events? If you have zero, and you want to divide it into N number of parts, well, you’d still have zero things, and so what? But what if you had somethings, and you wanted to divide them all into zero parts? Does that mean the operation represents the total annihilation of those things?

We can imagine a situation where total annihiliation by division is possible, but is it worth building that interpretation into most models, when an operation like that isn’t often applied to circumstances, and such an outcome is rarely requested?

My understanding is that the math professor interpreted the question to mean "what if our universe, rather than acting in so-far-seeming correspondence with ZFC set theory, was one that was best explained by a different axiomatic set theory—one that didn't entail the existence of "6" as a mathematical object? What universe would we be in, and what would it be like?"

> I thought this was going to be more interesting than 'imagine the world in base 6' - what're the ramifications of a 'hole' in the middle? I recall enough to know that fields allow for it, but are the consequences at all interesting?

Why? There's nothing interesting to say. If you have no 6, either you renamed it ( http://www.smbc-comics.com/index.php?id=3913 ) or you've stopped working in a useful structure.

In particular, ℤ without 6 is not an algebraic field, because fields are closed under addition and multiplication.

ℤ mod 5 is a useful structure where 6 doesn't exist.

6 is part of that structure, sharing space with 1 and 11.

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