wtf

Didn't read any of the other posts so apologies if someone else mentioned this. You can build logic gates out of dominoes. You can build logic gates out of anything. Transistors are convenient because they are very fast. But that's only an implementation detail. Nothing in the definition of computing requires any particular implementation; and in fact it's a basic principle of computing that the particular implementation is irrelevant. https://en.wikipedia.org/wiki/Domino_computer

I haven't the bandwidth today to respond in detail. I noted this particular phrase, which is the point at which you stopped working hard to understand the proof and instead substituted handwaving. First, the proof is a diagonalization, but it's not the Cantor diagonal argument. Secondly. "lead to paradoxes," no. It leads to the conclusion that TMH can not in fact exist. My suggested for you would be to go back to the proof of the unsolvability of the Halting problem; and no matter how long it takes you, go over ever single itty-bitty detail until you clearly understand the proof. Only then go forward with your new example, which frankly doesn't hold water at all. In short you are stopping well short of understanding right at the point where the work gets hard. You need to push through that and understand the proof.

You're basically reinventing linear algebra. You can form a matrix of the coefficients; and if the matrix is invertible, or equivalently has nonzero determinant, you can multiply right side of the equation (the constants) by the inverse of the coefficient matrix to get the answer. The benefit of this method is that it's easily programmed into a computer and requires no insight or cleverness as to what to do first. It's automatic. The answer to your question about the algebraic method is that you can generalize this by talking about "elementary row operations," and algorithms by which you apply them to reduce your system of equations so that it yields the answer. I Googled around and this looks like a pretty good overview of what you're asking about. https://en.wikipedia.org/wiki/System_of_linear_equations

I hope you don't mind a couple of very general remarks, since I'm profoundly ill-equipped to offer substantive comment on matters of physics. * Your title claims a disproof of something widely believed to be true. https://en.wikipedia.org/wiki/Bell's_theorem So the burden is on you to be clear in your exposition. Along these lines, it's not required but readers like myself would find it very helpful to have some context. Can you give us a little overview of the subject and perhaps an outline of why you think you have a disproof of something that's called, in the physics business, a theorem. A roadmap or high-level overview of why you think you know something that everyone else doesn't. * Nice pictures. As a math person I certainly appreciate the basic fractals like the Koch snowflake. I wonder if you can mention briefly how the study of mathematical fractals (which, after all, exist in continuous mathematical space) relate to physics, which, as I hope you know, resides in a space that's not known for sure to be continuous or discrete. * As a stylistic note, it's not necessary to quote your entire lengthy post every time you reply to yourself. All that does is cause people to have to scroll endlessly to read through the thread. Well anyway like I say I really can't help with the physics. But I have a lot of experience with people who claim to have disproved some famous theorem! The burden is on you to motivate readers to take you seriously. You can go a long way with context and clarity. ps -- I skimmed a little, found this: > You'll find that doing this will produce inside out spheres that are smaller than a Planck length: Ok first, what is an inside-out sphere? Earlier you said photons are spheres. Do you mean mathematical spheres? Is that true? I don't believe physics says that, but I'm no expert on photons. But i know spheres. What is an inside-out sphere? A sphere is just a set of points in three-space equidistant from the center. If you turned it inside out that would be meaningless, it would still be the same set of points. The sphere itself is a two-dimensional manifold embedded in 3-space. By the way you can reflect all of 3-space in the unit sphere, is that what you mean? Everything inside goes outside and vice versa, and the origin goes to infinity perhaps. Haven't thought about it. Another thing you can do is evert a sphere, turing it inside out by passing it through itself but (this is the clever bit) without leaving a crease. Surprising counterexample, nobody thought it was possible. https://en.wikipedia.org/wiki/Sphere_eversion So what exactly do you mean by turning a sphere inside out? Also, nobody can say ANYTHING sensible about anything smaller than a Planck length, because the Planck scale is the point at which physics breaks down. That doesn't mean there either is or isn't something interesting "down there," it just means that our present physics simply doesn't apply. We can't use to reason. So I found this one sentence to be troubling.

If I failed to see the distinction, what simple phrase or couple of sentences could you say to me that would give me a glimmer of an idea as to why you see a difference between G and f? G isn't related to Collatz, it's just some function you made up. That's how it seems to me. Can you articulate why you think these are different?

Why isn't Collatz a "natural byproduct" of f(x) = 2? Can you explain the difference in a few words? Why is G related to Collatz and f isn't?

@Zolar, Did you understand @taeo's excellent point? Let f(n) = 2 for all natural numbers 2. Then f satisfies your condition on G. It always outputs a power of 2, namely 2^1. Now given some number n, let's apply Collatz. If it's even, then eventually it either hits ("converges to" in your terminology") 1; or else it hits an odd number. If it's odd, then f(n) = 2. So f is "inside the Collatz function" in your concept. Yet I hope you can see that f is just some arbitrary function that has nothing to do with Collatz. It doesn't prove anything. Do you understand this example?

I simply can't help anymore. If I ever did. > For any natural number N, there exists a function G such that any N converges to a power of 2 over M iterations. So now there's a DIFFERENT G for each n?

It makes no sense at all to me. You've reverted to the "spinning out of control" narrative that I thought I talked you down from the past couple of days. I was mistaken. I did no good at all. I can't do anything else here.

I don't think I've served you by encouraging you in these last few pages. You haven't got a proof and your exposition is incoherent. I'm sorry.

Jeez man.

None of that is important. I understood what you meant by convergence very early on. You don't have to define what a theorem is. You have to figure out what you're trying to say about Collatz.

Yes but the collapsing to 1 is completely different in the two cases. You're convincing me you haven't a proof. In any event this convo is now meaningless unless you supply a proof of your assertions. You can't keep tossing out vague plausibility arguments, especially ones I don't consider plausible. You claimed a proof. You don't actually have a proof. I'd say that's a problem for your claim.

I didn't think about your idea in detail, but if that's your research program, you should definitely get to it. Take your time, work your program, report back. > 5: Since our function converges to 2^n , so does the odd collatz function Now that I understand the prime reduction function, I can't see at all how it relates to Collatz. It's a totally different procedure. You take a prime, add 1, drill each of its prime factors (to multiplicity) down to a power of 2. Well ok you have me believing that. If you have a reduction from 3n+1 to that, perhaps you should work on a very clear exposition of how this works. Because Collatz doesn't have anything to do with taking the prime factors of anything. I think there must be something wrong with your idea right here.

> I think it will be injective, i doubt it would be surjective. Of course it's not surjective, its output is restricted to powers of 2. > So I know there is a lot missing Correct. Burden's on you. > the odd collatz function will always converge at some point to a 2^n number, Of course. If you can prove that you become famous.

Ok. Is it important to you that the mapping from p to the final 2^n is injective? Meaning that two different p's give different n's? I don't think that's true but I haven't looked for a counterexample and won't bother if it's not important. So, onward. How does this prove Collatz?

Ok we can table that for later. So we build the tree, all the terminal nodes are powers of 2, and we do "something" to those powers of 2. Now what. Please go slowly, one step at a time. Short and clear please.

Ok so what happens at the end? You fill out the tree and you have a bunch of terminal nodes that are powers of 2. Do you add them up? Multiply them? Does it matter? Once we nail this down we have a complete description of G and we can move on.

I edited it a little, give it another glance if you would. The whole subject of what functions are computable by algorithms and which aren't is very interesting. So please be clear. Is my tree concept sufficient for your needs? Because it seems damn simple to me, which means that I understand it. And I like ideas I understand better than ones I don't! But if it's not sufficient for your purposes, do let me know.

There IS a function that produces exactly the primes, in order. It's the function f that maps the natural number n to the n-th prime p_n. This function exists. There's an algorithm for it: For each natural number n: If n is prime, print it, otherwise don't. It's slow but we don't care about that, all we care is that the function exists. It can be coded as an algorithm, that is, as a Turing machine. As such, it inputs a natural number and outputs the corresponding prime. Note that we'll need a subroutine that determines if a number is prime. For that we just use trial divisors. Also note that if we just want, say, f(45545434343), we just run it that far and print out only the value we want. There is no question that such a function exists, that it's computable, and that a brand new programming student would be able to write within a few weeks. The problem is, we don't have a nice simple closed-form expression for it. We have to brute-force it. So the question isn't the existence of such a function. It's that we don't have an expression for it.

The claim that every branch terminates in a power of 2 is obviously true but a formal proof eludes me. I haven't spent much time on it. Something about the induction bothers me but I can't think of how a counterexample would work. I've decided to put the issue aside for now. So let's just say I believe every subtree terminates in a power of 2. But this is different than your "adding" idea. Is it sufficient to get your argument off the ground? Either you have to accept that my "tree" method serves your purposes; or else you have to explain to me your adding idea, which I confess makes my eyes glaze.

> I will renew my efforts to clearly define a set of functions No, please, let's go back to the point where we were in agreement and work from there. I have a plan. We were in perfect agreement at a certain point then (I feel) you spun off in a million different directions. The expression that came to my mind was "thrashing," which is what a computer does when it's so busy managing virtual memory that it can't get any actual work done. It gets busier and busier but accomplishes nothing. Please don't take that as pejorative or "condescending." God it's not the first time I've been called that online. It's just the way I write. Or as Jessica Rabbit said, "I'm not bad, it's just the way I'm drawn." Believe it or not I'm trying to be helpful with those remarks. Because we're having TWO separate conversations. One is about the math. The other is about the exposition of the math. So I am not attacking your idea. I perfectly well stipulate that you might be Ramanujan and I'm damn well not Hardy. Maybe you're right that a smarter person than me could figure out what you're talking about. But on exposition, I am simply playing the part of a typical, somewhat dim, reader of mathematics. I read a line, try to figure out it, write down some simple examples. So when you write some exposition and I say you're thrashing, I am simply telling you that you have lost your reader. You have to make it much simpler to get through to the likes of me. Now here is what I posted before. Speaking of mathematical communication, we ALL have a lot to learn. I thought I was making a clear point, a meta-point, a big-picture point. But I forgot to make my point! So here it is. There isn't a convenient "function" for G. Just express it as an algorithm. For that matter, this is exactly how Collatz works! I can't think of a convenient way to express Collatz as a function, even though there is a function in theory. But in practice it's far better to work with it as a procedure or recipe. So if you would forget about calling G a function, and notating it as if it were a function, your exposition would improve dramatically and I wouldn't have to spend so much time yelling at you that G's not properly defined. I'm sure we're both tired of it by now. G is not a function. It's a procedure. Think about it that way and talk about it that way. Else we're doomed. Having said that, we are in agreement through step 3. You did at one point say that you "keep the power of 2 and only divide it out at the end." That is the first moment of murky divergence from our point of agreement. Not only that, in your recent posts you are clearly using the 2^n in some way along with the other primes. So this is important to you and I don't quite understand it. It seems to complicate things. Now ignoring the 2^n, whether you divide it out or include it in something later, we have the remaining primes. For example in the case of G(139), we have 139 + 1 = 140 = 2^2 * 5 * 7. Note that G(139) is our notation for "Input the prime 139 into the G recipe." It is not a function. But now that we've defined the notation, it's ok. As long as we only input primes!! Now my idea, which is evidently not what you have in mind, is then to recurse through the procedure on each of 5 and 7. So 5 + 1 = 6 = 2 x 3 and 3 + 1 = 2^2. And 7 + 1 = 8 = 2^3. So we claim (although I have not yet formally proved) that each subtree terminates at a power of 2. But that is my idea. (*) At that point, you tried to explain to me that you do something with five 7's, and that's where I lost the thread. And in an attempt to explain this to me, you're (FROM MY VIEWPOINT, this is not meant to be pejorative) spiraling out of control. So if you see any hope of going back to our most recent point of perfect agreement, that might be helpful. (*) I'm nervous about this. I can see it's a strong induction ... having proved it for all primes below a certain point, it's true for larger ones because p+1 has factors that are smaller than p. On the other hand, it's a funny induction. The induction step never depends on any of the values that have gone before. I am starting to wonder if it's true. For example ... suppose we know that 3, 5, 7, 11 all go to 2^n. What if we then put in a p such that p+1 only has prime factors of 13 and above? There's no actual induction step, you have to look at every prime by hand. I am starting to wonder if there might in fact be a counterexample. Your argument depends on p_i < p so I believe this impacts your argument.

So by definition, G(103 + ... + 103) isn't defined because the argument to the function is not prime. > p = 2^n x 7 x 7 x 7 x 101 x 101 x 103 x 103 x 103 x 103 No, that number is not prime. 'p' has denoted a prime since the beginning. If that's no longer true, you sure didn't mention it. I think instead of posting anymore, you should go back to the drawing board and try to work through your idea. Maybe someone else can offer suggestions. I'm out of ammo.

I can't make heads or tails. I can't be helpful anymore. I can see you're multiplying and adding a bunch of numbers, but there doesn't seem to be anything for me to work with. I need to let this go. On the left hand side you have G of a composite number. We're already in perfect agreement that this is not defined. You can input a prime to G and recurse down a bit till your exposition falls apart, but you can NOT input a composite number to G. I have to let this go, I have nowhere to go with this.

> - I would hope so, Ive been sitting on this for almost a year. Numerous scatalogical jokes omitted. > extract, to take out one piece of a whole. Think about a number as an entire collection of collections of objects that are equivalent. I am certain I have no conceivable idea what you are talking about, even after imagining various set-theoretic constructions of various classes of numbers. For the third or fourth time in a row you are pointedly ignoring my request for context. > Currently we select a prime and reconfigure the product of primes into a sum of that prime. Then iterating through that on each prime. "reconfigure." What am I supposed to make of that? You still won't tell me how to iterate through 5 x 7. I don't want to sound negative, but your last half dozen posts haven't made any sense and haven't addressed my concerns, nor to you seem to be making any progress in clarity. I am near the end of my ability to be helpful. I did some useful work in helping you to define G, but once you do the initial factorization of p+1, the rest of your idea is simply unexplained. I think we're at around three days with no progress at all. > The core concept of adding 1 to a prime and reducing the primes within it is the primary mechanism we want to chase. On a very rudimentary level, the mathematics that I know, only allow you to do something like add to a particular value only if you can group it together. Obviously you cannot just select random primes within a product to add 1, but you can do just that over a sum of primes. Simply doesn't parse. Maybe I should stop posting. I have nothing to add.