## Trying to find a way to describe this "set" of numbers

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zenten
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### Trying to find a way to describe this "set" of numbers

OK, so I came up with this germ of a thought in my head a few months ago, and I can't get it out. I want to either figure out how to formally define and describe for it for others, or come to the conclusion that what I'm thinking of is incoherent.

So, here's the idea. There's a countably infinite "set" (I am unsure if it fits the formal definition of sets) of "numbers" (and I don't know if they fit the formal definition of "number"). Now, in many ways these numbers behave like the Real numbers (if I can find a way to extend the definition to a complex number analogy later I'll work on that too). But there is one *huge* difference. Every number is unique.

Now, here's what that means. Lets suppose A and B are examples of these numbers. So saying A = B is not a valid statement (or possibly is just always false, I'm not sure). You could however state A > B or A < B, and one would always be true and the other would always be false. Also, statements like A > B + 2 would also be invalid if A and B, but you could do things like A + B > 2. Also, in any given statement each variable representing one of these numbers can only appear once, so you can't say things like A + A > B.

Otherwise where possible this set would have the same properties as Real Numbers. So the = comparison doesn't work (or again, maybe is just always false) but things like multiplication and exponents and division etc do. I'm also thinking that the absolute value function would not work, nor would any other function that can take two or more inputs in this set and output only one value.

So, how about it, does this concept make any sense? If so, how would I go about formally defining it?

gmalivuk
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### Re: Trying to find a way to describe this "set" of numbers

If you can't say A=B, then I'm not sure how your construction makes any sense.
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zenten
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### Re: Trying to find a way to describe this "set" of numbers

gmalivuk wrote:If you can't say A=B, then I'm not sure how your construction makes any sense.

OK, I'm not sure that you can't exactly.

What I *do* know for sure is that you can't say the following.

Let A, B, C, D and E be unique examples of these numbers.

If A + B = C, then D + E != C for all A, B, C, D and E.

If A - B = C, then D - E != C for all A, B, C, D and E.

If A * B = C, then D * E != C for all A, B, C, D and E.

If A / B = C, then D / E != C for all A, B, C, D and E.

Also, no arithmetic operation (and I totally get that say + in these kind of numbers would have to be subtly different than what is used in Real numbers) containing only numbers in the type that I'm describing can result in an integer.

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### Re: Trying to find a way to describe this "set" of numbers

Equality is more fundamental than arithmetic, its necessary for sets, and the elements of sets, to make sense.

In particular, in you numbers you talk about "2". Well, if, when I write "2", I always mean the same thing, then... it doesn't really make sense to say that "2=2" is false, or an invalid sentence. Its fundamental, and necessary, that we be allowed to say that and that it be true.

I'd be interested to try to understand what inspired these ideas. There is a thing called the computable numbers, and equality is not a computable relationship, but ">" and "<" are.

Edit to add: according to your new post, you've pretty much stomped all over manipulating equations. I'm not sure what purpose these numbers could have.
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### Re: Trying to find a way to describe this "set" of numbers

I'm also not clear what new definition you're giving to the binary operations if none of those equalities can ever hold.
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zenten
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### Re: Trying to find a way to describe this "set" of numbers

jestingrabbit wrote:Equality is more fundamental than arithmetic, its necessary for sets, and the elements of sets, to make sense.

In particular, in you numbers you talk about "2". Well, if, when I write "2", I always mean the same thing, then... it doesn't really make sense to say that "2=2" is false, or an invalid sentence. Its fundamental, and necessary, that we be allowed to say that and that it be true.

I'd be interested to try to understand what inspired these ideas. There is a thing called the computable numbers, and equality is not a computable relationship, but ">" and "<" are.

Edit to add: according to your new post, you've pretty much stomped all over manipulating equations. I'm not sure what purpose these numbers could have.

I got the idea after reading Feynman's QED: The Strange Theory of Light and Matter. I was reading his description on the issues with using renormalization and I was wondering if using a different kind of numbers. I don't know if my idea would work or not, but I do know that coming up with a formal definition of my idea would be needed first.

I realize that it doesn't follow set theory. Which is fine, I'm trying to come up with a method for doing arithmetic, not work on sets. It's why I put "set" in quotes in my first post.

Let me try to explain this from another angle. Lets assume that the universe is continuous (which I realized is a big assumption). Assuming you are only alive a finite amount of time (less of a huge assumption) this means every quantity you measure will actually have a slightly different value, even if more than one measurement might be approximated to the same value as another. Every length, every mass, every duration will actually be slightly different. And because of relativity your reference frame is never constant, so even if you are somehow measuring the exact same thing twice from your reference frame it's always going to be a slightly different value. I'm trying to think of a way of doing arithmetic that takes that into account.

And anyway, math doesn't require everything to have a purpose. Even if this turns out to not work for what I'm trying to do there's nothing to say that someone else might not find it interesting. And at the very least I'll learn something here by making this attempt. I already have learned a lot by looking into things that ultimately don't work for this.

Demki
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### Re: Trying to find a way to describe this "set" of numbers

zenten wrote:
jestingrabbit wrote:Equality is more fundamental than arithmetic, its necessary for sets, and the elements of sets, to make sense.

In particular, in you numbers you talk about "2". Well, if, when I write "2", I always mean the same thing, then... it doesn't really make sense to say that "2=2" is false, or an invalid sentence. Its fundamental, and necessary, that we be allowed to say that and that it be true.

I'd be interested to try to understand what inspired these ideas. There is a thing called the computable numbers, and equality is not a computable relationship, but ">" and "<" are.

Edit to add: according to your new post, you've pretty much stomped all over manipulating equations. I'm not sure what purpose these numbers could have.

Let me try to explain this from another angle. Lets assume that the universe is continuous (which I realized is a big assumption). Assuming you are only alive a finite amount of time (less of a huge assumption) this means every quantity you measure will actually have a slightly different value, even if more than one measurement might be approximated to the same value as another. Every length, every mass, every duration will actually be slightly different. And because of relativity your reference frame is never constant, so even if you are somehow measuring the exact same thing twice from your reference frame it's always going to be a slightly different value. I'm trying to think of a way of doing arithmetic that takes that into account.

Seems like you're looking for Wave Functions or Interval Arithmetic.
Also, "slightly different value" different from what?

zenten
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### Re: Trying to find a way to describe this "set" of numbers

Demki wrote:Seems like you're looking for Wave Functions or Interval Arithmetic.
Also, "slightly different value" different from what?

Hmmm... no, doesn't quite fit what I'm looking for either.

By "different" I mean different from the last time you measured. No two reference frames can ever be identical given a continuous universe.

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### Re: Trying to find a way to describe this "set" of numbers

But even if there are no identical measurements (which is false, because sometimes we want to measure integers), there are still identical values ie the speed of light, the charge on an electron, the half life of a proton etc etc.
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### Re: Trying to find a way to describe this "set" of numbers

I think the OP is trying to come up with an operator and set combination (say (C, *) for some countable set C) where each application of * produces a new member of C. That is, each member of C can be uniquely represented ('factorised') by a finite string of elements of C, which when *ed together, get that element.

Now, it's apparent that * can't be associative, and it's probably easier if it's not commutative either.

So let's start out with those as axioms (I'm being pretty non-rigorous with saying what sets things come from, but it should be pretty obvious):

1) If A1 * A2 * ... * An = C, and B1 * B2 * ... * Bm = C, then m = n and Ai = Bi for each i.
2) A > B iff there exists C1, C2, ..., Cn such that:
B * C1 * C2 * ... * Cn = A, or
C1 * C2 * ... * Cn * B = A, or
C1 * ... * Ci * B * Ci+1 ... * Cn = A
3) if A1, A2, ..., An are in C, then A1 * A2 * ... * An is in C

Are there any obvious corollaries? Contradictions? Any extra axioms you can add to make it more like real arithmetic? Can such a C be generated from one element? Can C be countable with these axioms?

zenten wrote:By "different" I mean different from the last time you measured. No two reference frames can ever be identical given a continuous universe.

In a uniform, continuous universe, all reference frames are identical.

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### Re: Trying to find a way to describe this "set" of numbers

Even if all measurements are different, we might want to talk about the same measurement twice. E.g., if I take measurement A, and then I take measurement B, and then I compute A + B and B + A (using the same measurements, not taking new ones), I'd expect to get equal results for those computations, even if I couldn't get the same result if I took the measurements again.

dudiobugtron wrote:I think the OP is trying to come up with an operator and set combination (say (C, *) for some countable set C) where each application of * produces a new member of C. That is, each member of C can be uniquely represented ('factorised') by a finite string of elements of C, which when *ed together, get that element.
The set of all expressions?
If we start with a countable number of elements that can't be produced by *, then C is countable, since each element in C can be represented by a string, and the set of all strings is countable.
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### Re: Trying to find a way to describe this "set" of numbers

dudiobugtron wrote:Now, it's apparent that * can't be associative
...
A1 * A2 * ... * An

Please don't write multiple applications of a non-associative operation like this. You're confusing the hell out of me.

I thought I found a contradiction in your axioms. Then I remembered * was not supposed to be associative. Now I don't know what your axioms even mean.
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### Re: Trying to find a way to describe this "set" of numbers

skeptical scientist wrote:
dudiobugtron wrote:Now, it's apparent that * can't be associative
...
A1 * A2 * ... * An

Please don't write multiple applications of a non-associative operation like this. You're confusing the hell out of me.

My apologies, this was pretty haphazard of me. I was just assuming the operations would be processed from left to right. So by A*B*C, I mean (A*B)*C.

Note, though, that regardless of the order you do the *s in, as long as it is consistent then it shouldn't really change the axioms at all (they just might need to be slightly rephrased if the order is really weird).
Also, it's really hard to use the brackets clearly in lists with ellipsis, since the number of brackets you need at the start depends on n.

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### Re: Trying to find a way to describe this "set" of numbers

I just took a look at your axioms again, and noticed that in axiom 1) you insist that m=n. This is inconsistent with axiom 3, because A * B = C for some A, B, and C, but 2≠1.

Here's a modified version of your axioms (rewritten to avoid offending my aesthetics) which is consistent:
dudiobugtron' wrote:1') If A * B = C * D, then A = C and B = D.
2') A > B iff there exists an associated product of elements of C equaling A, with at least 2 terms, where one of the terms is B.
3') if A and B are in C, then A * B is in C

Axioms 1' and 3' are much simpler than your axioms 1 and 3 because I only consider binary products, but they imply your axioms 1 (minus the restriction that m=n) and 3, using induction. Axiom 2' is more general than your axiom 2, because I consider arbitrary ways of associating the product C1 * ... * Ci * B * Ci+1 ... * Cn (which makes more sense, imo). It's kind of awkwardly stated, so it would be nice if someone else has a cleaner restatement of the axiom.

These axioms are consistent, because they are modeled by the Dyck language, with A * B = [A][B] and A < B defined as in axiom 2'. (As < is not mentioned anywhere else, it won't break anything to take that axiom as the definition. I really wanted to define it as some variant of substring, but sadly this does not work.)
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### Re: Trying to find a way to describe this "set" of numbers

skeptical scientist wrote:I just took a look at your axioms again, and noticed that in axiom 1) you insist that m=n. This is inconsistent with axiom 3, because A * B = C for some A, B, and C, but 2≠1.

Well pointed out. The problem with my definition is that I was trying to set up a situation where A * B = C, but A * B * D != C * D, which is pretty non-sensical.
I think this was sort of what the OP was going for, so I don't think my axioms (or your better, modified axioms) fit the OP's purpose.

However, I am not the OP, so I might be wrong!

Anyway, I think you could re-write your Axiom 2 in a 'recursive' way, like this:
A > B iff (B * C = A or C * B = A for some C) or (A > D and D > B for some D).

Not sure if that's better though, nor particularly nicely written.

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### Re: Trying to find a way to describe this "set" of numbers

dudiobugtron wrote:I was trying to set up a situation where A * B = C, but A * B * D != C * D, which is pretty non-sensical.
I think this was sort of what the OP was going for, so I don't think my axioms (or your better, modified axioms) fit the OP's purpose.

Yeah, if you want A * B = C, but (A * B) * D ≠ C * D, you're violating one of the basic logical axioms of equality, so you're essentially redefining the equals symbol (which has a consistent meaning throughout all of mathematics). That was one of the things that bothered me about the OP—being able to refer to the same object more than once is such a basic part of mathematics that it's hard to formalize a system where that's not the case.

Anyway, I think you could re-write your Axiom 2 in a 'recursive' way, like this:
A > B iff (B * C = A or C * B = A for some C) or (A > D and D > B for some D).

I thought of that, but unfortunately it's not logically equivalent.
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### Re: Trying to find a way to describe this "set" of numbers

skeptical scientist wrote:
Anyway, I think you could re-write your Axiom 2 in a 'recursive' way, like this:
A > B iff (B * C = A or C * B = A for some C) or (A > D and D > B for some D).

I thought of that, but unfortunately it's not logically equivalent.

It's not? I'm interested in seeing a counter-example, since I can't think of one at the moment

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### Re: Trying to find a way to describe this "set" of numbers

dudiobugtron wrote:
skeptical scientist wrote:
Anyway, I think you could re-write your Axiom 2 in a 'recursive' way, like this:
A > B iff (B * C = A or C * B = A for some C) or (A > D and D > B for some D).

I thought of that, but unfortunately it's not logically equivalent.

It's not? I'm interested in seeing a counter-example, since I can't think of one at the moment

If A>B holds for all pairs (A,B), then your version of the axiom holds.
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### Re: Trying to find a way to describe this "set" of numbers

So does yours, though; although I guess in that case, yours also shows that for each A, each B is in some associated product of elements of C equaling A with at least 2 terms, whereas mine doesn't show that there is some D less than A with B * C = D or C * B = D for some C. So perhaps I should reword it to:

A > B iff (B * C = A or C * B = A for some C) or (for some D, A > D and B * C = D or C * B = D for some C)

Maybe there should be another axiom which prevents situations where A>B and B>A?

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### Re: Trying to find a way to describe this "set" of numbers

dudiobugtron wrote:So does yours, though;

Not necessarily. It could be, depending on the structure of C under *, but it could be false. Yours is necessarily true if A<B always holds.

A > B iff (B * C = A or C * B = A for some C) or (for some D, A > D and B * C = D or C * B = D for some C)

This has the same problem. If A>B holds for all pairs A,B holds, then A>B*C will hold for all triples A,B,C, and so your axiom will be true.
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### Re: Trying to find a way to describe this "set" of numbers

Going by the premise that for unique elements a, b, c, d, and e, if a+b=c then d+e != c, and further that a+d != c, but that a=a is still valid.

We can construct some objects that follow this rule.

1) '0' is a member of our set.
2) Given any not-necessarily-unique members a and b, [a,b] is a member of our set, where [a,b] is an un-ordered pair.

Define ordering by the rules below.

If b is an element of a, then a>b, and vice-versa.
Else, denote the greater member of a by g(a): g(a)>g(b) implies a>b, and vice-versa (g(a) is an arbitrary selection if both members are the same).
Else, denote the lesser member of a by l(a): l(a)>l(b) imples a>b, and vice-versa.
Else, a=b.

So the first few members of the set are:
0, [0,0], [0,[0,0]], [[0,0],[0,0]], [0,[0,[0,0]]], and so on.

I haven't worked out a definition of multiplication for which every possible multiplication has a different answer.
EDIT: I suppose you could use the ordering to associate each of these numbers with a natural number, then define multiplication the usual way as repeated addition.

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### Re: Trying to find a way to describe this "set" of numbers

That's very similar to the definition dudiobugtron and I were playing with. If you want to define addition and multiplication to both have those properties I don't think it's possible to also get distributivity.
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### Re: Trying to find a way to describe this "set" of numbers

skeptical scientist wrote:
dudiobugtron wrote:Now, it's apparent that * can't be associative
...
A1 * A2 * ... * An

Please don't write multiple applications of a non-associative operation like this. You're confusing the hell out of me.

I thought I found a contradiction in your axioms. Then I remembered * was not supposed to be associative. Now I don't know what your axioms even mean.

We do this with -, /, and ^ already. All you need is a convention whether to read it left or right associatively.

In any case, it sounds like we're just looking for the free magma on a free magma (one free for each operation). It's a pretty unwieldly structure, but there are probably some trivial total orders you can put on them to get the properties you're looking for.
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### Re: Trying to find a way to describe this "set" of numbers

The only times I've seen it done with / is when trolling, because there isn't such a convention and so the result is ambiguous.
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### Re: Trying to find a way to describe this "set" of numbers

Every device and/or program in my house or on my computer treats / as left associative, including 4 calculators, my iphone, excel, C, python, haskell and W|A. It's also implied by http://en.wikipedia.org/wiki/Order_of_operations, as treating /3 as * (1/3) makes it left associative.

Certainly when I teach students order of operations, I emphasize that multiplication and division together are to be done left to right, just as addition and subtraction are.

In any case, on the main issue, defining < based on being able to 'get to' something via operations is only going to give a partial order. If you put an order on the 'generators', you can get a lexicographic ordering on the set generated by 1 operation, but I'm not sure that you can get a total ordering with 2 operations.
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