wtf

Strings talking about sets are not sets. Well the set of primes is a set. But the string "n in N such that n is prime" is not a set! I agree that the string has a Gödel number, and you can assert that collections of Gödel numbers are sets, but I don't think you can ask if "n in N such that n is prime" is an element of the set of primes. Well yes all the statements have Gödel numbers but I don't think it works this way. I'm out of my depth on Gödel numbering here so maybe you can explain why you think it works this way. I hope you get the million dollars. I confess that I'm annoyed at myself that I can't say exactly why your argument is wrong. If only I'd worked harder studying Nagel and Newman back in the day.

Looking at your paper, I see some issues. 1. What is a "set definition?" Do you mean a predicate? But predicates don't define sets, that's Russell's paradox. So what do you mean by a set definition? 2. You have the expression [math]X \notin S(X)[/math]. But [math]X[/math] is a "set definition," whatever that is, and NOT a set; and [math]S(X)[/math] is a set. So your expression makes no sense, it's neither true nor false. 3. Likewise [math]M \in S(M)[/math], has the same problem. [math]M[/math] is a set definition and not a set; and [math]S(M)[/math] is a set. Perhaps you can clarify, starting with what you mean by a "set definition." Or perhaps a set definition is a specification like [math]\{n \in \mathbb N : n \text{ is prime} \}[/math]. But in that case a set definition is a piece of syntax, a string in the formal language. It can't be an element of some set that the language is talking about. You're mixing syntax with semantics, formal strings with models. It's a plot by the publishers to sell more books. Or: Have you seen viXra lately?

This popped up on my radar this morning. "Finding primes using a parabola." Looks like it might be of interest to you. https://www.cantorsparadise.com/finding-primes-using-a-parabola-a939c6e8be53 First time I clicked on it the story appeared, then after that it said I needed a paid app. Not sure how this works but perhaps you'll be able to view it.

You have not demonstrated this relationship for 5, 17, and 85. The best you can do is compute the sector area for 5 and 17 and then adjust the spiral constants after the fact. You'd then have to pick different constants for every pair of numbers. And you could do exactly the same for 4, 6, 24; or 8,10,80. First, the primality of the first two numbers isn't needed; and secondly, for each pair of radial lengths you need to adjust your spiral constant after the fact. Don't you see this? What are the spiral constants for 5 and 17?

It doesn't have any truth OR any falsity. You haven't stated any clear proposition that would have a definite truth value. You haven't said why radial lengths of 5 and 17 are meaningful. I could do the same exercise with lengths of 4 and 6. But I don't mean to argue with you. I learned a little about the log spiral and perhaps helped you to clarify some of your thinking, or not, as the case may be.

That's exactly what I did earlier when I computed the sector area between radial segments of lengths 5 and 17. That's when you asked me why I didn't account for the angle. It's because the formula doesn't involve the angle, because given the radial lengths, you can compute the angle. That's this post: https://www.scienceforums.net/topic/124453-simple-yet-interesting/?do=findComment&comment=1178970 Oh I see what you mean. I interpreted [math]r(\varphi)^2[/math] as functional notation. [math]r(\varphi)^2[/math] is the square of the radius associated with the angle. I didn't take it as multiplication. It doesn't make any sense to square an angle. I see your point, the notation's a little ambiguous but I think they mean to input the angle to obtain the radius, then square that radius. I could be wrong. Either way it doesn't matter. There's nothing about primes or semiprimes about any of this. You pick two angles and get a pair of radial lengths and compute the sector area, but it doesn't mean anything with regard to primes or semiprimes.

The primality of 5 and 17 have nothing to do with anything, nor have you reached or stated any conclusion. You take any two angles and you will have radial segments with a sector area between them as computed by the formula on Wiki. It doesn't seem to mean anything. What happened to the radial segment of length 5? What point on the spiral corresponds to a radius of length 17? If the spiral constants are [math]a = 1, k = 1[/math], the spiral is given by [math]r = e^\varphi[/math]. What value of [math]\varphi[/math] gives a radius of 17? [math]\varphi = \log 17[/math], right? That's the angle, in radians. It comes out to around 161 degrees. [math]\log 17 \approx 2.8][/math], and 2.8 radians is around 160 degrees. That's the angle it would make with the line from the origin to (1,0) if you plug in [math]\varphi = 0[/math]. How are you making your calculation?

It's the sector area formula from wiki. https://en.wikipedia.org/wiki/Logarithmic_spiral#Properties

17 is supposed to be a radial length. What is 0 here? Can you clarify? What you wrote doesn't seem to make sense in the context of the problem. Did you mean the angle between radial lengths of 5 and 17? Well if the spiral is [math]r = e^\varphi[/math] then the angles are [math]\log 5[/math] and [math]\log 17[/math], respectively, [math]\log[/math] meaning natural log. Is that what you're doing? From the sector area formula on Wiki, the sector area between radial segments of length log 5 and log 17 is (17^2 - 5^2)/4 = 264/4 = 66. That's with spiral constants a = 1 and k = 1. You could adjust k to make the area come out to 85, but I'm not sure what point is being made. Have you looked at the Wiki page on the log spiral? Your log spiral changed into a parabola? As Galois said as he was frantically writing up his mathematical discoveries the night before he was to die in a duel, "I have no time. I have no time."

Ok referring to https://en.wikipedia.org/wiki/Logarithmic_spiral the log spiral is given by [math]r = a e^{k \varphi}[/math] where [math]a > 0[/math] and [math]k \neq 0[/math] are real constants. The sector area is [math]A = \frac{r(\varphi_2)^2 - r(\varphi_1)^2}{4 k}[/math] where [math]\varphi_1[/math] and [math]\varphi_2[/math] are your two angles. So: What values of [math]a[/math] and [math]k[/math] have you chosen in order to get radial distances of 5 and 17; and what are the corresponding angles; and does the sector area work out to 85? You should work this out and see what makes sense. You need to find the constants and the angles that make your picture valid. And the primality of the radial lengths doesn't appear to be significant in any way, so can you explain why you think it is? If there's anyone here who knows anything about spirals, I'm confused about something. In the picture on the Wiki page, they show the sector area as going between the origin and two points on the spiral. But clearly the radial lines pass over other parts of the spiral more near to the origin. What defines the sector, exactly? How do you know which arm of the spiral defines the sector? I don't know anything about spirals, I'm confused by this. ps -- Oh I see what's going on. The radial line segment is defined from the origin to the point [math](r, \varphi)[/math]. So for example take a log spiral with [math]a = 1[/math] and [math]k = 1[/math], so the spiral is given by [math]r = e^\varphi[/math]. Then for an angle of [math]0[/math], the point on the spiral is [math](1,0)[/math] in polar coords. If you go around the circle again with [math]\varphi = 2 \pi[/math], the corresponding point is [math](e^{2 \pi}, 2 \pi) \approx (535, 0) [/math]. In other words the angle is always taken mod 2 pi, but the radius keeps growing (or shrinking towards zero as the angle decreases to minus infinity). So if two angles are close together (within 2 pi of each other) the sector makes sense; but if not, it's unclear what the sector is. Anyway I'm beginning to get some feel for the log spiral. "Today I learned!"

Well if the angle is zero, the area is zero. And if the angle is 180 degrees, the area is infinite, depending on how you calculate the area, because you'd be crossing the spiral multiple times. Maybe you only count the area till the first crossing, in which case it's finite. So there's probably some angle such that the area is exactly 85, unless you only count to the first crossing, in which case there might not be. But so what? The same can be said for any two radial lengths whatsoever. I don't see why this is meaningful. Explain please what the primality of the lengths has to do with anything.

Ok I believe you if that's true. What's special about primes? Wouldn't it just depend on the properties of the log spiral? You could just as easily use 4, 6, and 24, if that multiplicative property is true. But even though this isn't drawn to scale, clearly the distances of 5 and 17 are wildly off. Are you sure any of this is accurate? Are those supposed to be distances from the center? Or arc length along the spiral? The formula for sector area is given on the wiki page. https://en.wikipedia.org/wiki/Logarithmic_spiral#Properties I find it somewhat unlikely that all these would be integers, given the formula for sector area. I haven't worked any of this out, I'm wondering if you have and if any of this is actually true.

The log spiral has many interesting properties. For example, "the distances between the turnings of a logarithmic spiral increase in geometric progression." Perhaps this relates to something you're interested in. https://en.wikipedia.org/wiki/Logarithmic_spiral Can you draw a sketch of how 5, 17, and 85 relate to the log spiral? I think this might make your idea more clear.

Why is that? Isn't there a right triangle with sides [math]5, 17[/math], [math]\sqrt{5^2 + 17^2}[/math] ? Aren't there lots of other triangles with those two sides? The angle between those two sides can be absolutely anything strictly between 0 and 180 degrees. Can you try to explain what constraints you are trying to put on the angle, and what they are based on? Remember, coordinate systems are not inherent in geometry. The geometry comes first, and then you impose a coordinate system. The coordinate system can never change the underlying geometry, it can only make the equations simpler or more convenient in some cases. But the shapes, the angles and lengths, don't change. You could interpret plane vectors as complex numbers and divide them that way. I'm not sure what this line of thinking would do for the OP, who is having a hard time articulating his idea. I don't think anyone would ever take the number 31, say, and note that "Hey that factors as the dot product of (2,3) and (5,7)." Pretty wild thought ... Wonder if it means anything. Given any integer we can form the set of all the pairs of vectors whose dot product is that integer. This can't possibly amount to anything, else we'd have heard of it, especially in physics in the context of Hilbert space inner products. There must be some geometry in here somewhere ...

I would still like to see a simple, complete worked example with lengths 2 and 3. Put the 3-side along the positive x-axis with one end at the origin; and show me exactly how you place the 2-side such that the angle is somehow uniquely determined by ... something.

When they hacked the east coast pipeline I said nothing, because I live on the west coast. And by the time they hacked the US meat supply, there were no cows left to speak up for me! or something like that. https://www.washingtonpost.com/business/2021/06/01/jbs-cyberattack-meat-supply-chain/

This statement is false on its face. That's why readers are put off and have no idea what you're talking about. Could you give a specific example? Say I have a line segment of length 2 and another of length 3, both with one end at the origin of the x-y plane. Can you show how you get the third side? What additional information is supplied? Please be specific. No need for extraneous discussions of the travails of your long-suffering teacher. Just show an example of two lengths and how you determine the third one. Also, what does pnp stand for? Give a specific, completely worked out example and leave out all other extraneous commentary, and that will go a long way toward clear mathematical communication. Of course if you have two sides of a triangle plus some as yet to be specified information, the additional information might be sufficient to work out the third side. Why don't you give an example so people can see what you are talking about?

It's not just that China as a whole controls the vast majority of the mining. It's also that the top four Chinese mining pools control over 60% of the bitcoin hashrate, and have done so ever since I started tracking it several years ago. https://btc.com/stats/pool Basically, the conditions for a so-called 51% attack already exist and have always existed. These mining pools can wake up any morning they like and steal everyone's bitcoin. The only reason it hasn't happened yet is that they haven't done it. Not because they can't. ps -- Just to clarify what a 51% attack is, the idea is that the bitcoin protocol works by consensus. If most miners verify a block, the the block is good. If a bad actor inserts a bad block (containing, say, a transaction to transfer a thousand of your bitcoins to me), the majority of legitimate miners will reject that block and the bad transaction will fail. Now if bad actors take over more than half of the hashrate, they can collude to verify bad blocks and verify fraudulent transactions. Satoshi's original idea was that mining would be done by millions of individuals running PCs in their spare bedrooms, making collusion difficult and control of more than half of the network by bad actors, impossible. What Satoshi did not foresee was the processor arms race in which individuals with PCs have no hope of competing with huge miners who own highly specialized mining rigs. Only the big players can mine bitcoin now, and this has led to centralization. Remember that the entire mythos of bitcoin is that there is no central control. Well as it's turned out in real life, bitcoin is highly centralized after all, and collusion among bad actors is a real possibility. Finally I should mention that mining pools are associations of many independent miners, and it's not clear what kind of centralized control the operators of a pool have over their individual miner participants. So things are not quite as dire as I'm describing, but far more dire than you'd know if you only listened to the claims that bitcoin is decentralized. The publicly available hashrate distribution statistics bear that out.

A topological space does not necessarily have a metric. The definition of space is far more general.

Yes. The energy cost is the cost of validating blocks. That never changes. Even after the limit of creating bitcoins is reached, blocks must still be validated whenever there is a transaction. A good question is, why would the miners keep mining when they can no longer create new bitcoins? According to Investopedia, they would still collect transaction fees. https://www.investopedia.com/tech/what-happens-bitcoin-after-21-million-mined/

Thanks much. This is the answer to swanstont's question. If (in the extreme case, to illustrate the point) it takes no energy to verify a block, a bad actor can just verify false blocks at will and steal everyone's money.

No that's not possible. All proof-of-work based cryptocurrencies rely on the enormous expenditure of computational energy to verify the transactions. If verifying blocks is cheap, the protocol doesn't work. There are some efforts to mitigate the problem, such as green crypto mining. But one argument for the eventual failure of bitcoin is that it's out of spirit with the age. Bitcoin wastes energy extravagantly; and we live in the age of conservation. Not endorsing this service, just a link I googled randomly to illustrate that people are thinking about the issue. https://www.perpetualindustries.com/gem-cryptomining.html Another alternative is proof-of-stake, which is more energy efficient than proof-of-work. https://www.investopedia.com/terms/p/proof-stake-pos.asp

[math](-1)(-1) - 1 = (-1)(-1) + (-1) = (-1)(-1) + (-1)(1)[/math] [math]= (-1)(-1 + 1) = (-1)(0) = 0[/math] So that [math](-1)(-1) - 1 = 0[/math] and therefore [math](-1)(-1) = 1[/math]. Then [math](-3)(-2) = (3)(-1)(2)(-1) = (3)(2)(-1)(-1) = (3)(2)(1) = 6[/math]. All this follows from the associative, distributive, and commutative properties of a ring, which are satisfied by the integers. https://en.wikipedia.org/wiki/Ring_(mathematics) http://www.maths.nuigalway.ie/MA416/section1-2.pdf

Yes thanks.

I looked back at your comments in this thread and perhaps I missed your question, can you please repeat it? The only question I saw was that you asked what is the additive inverse of a + bi; and it's of course -a =bi. If there was another question I did not see it.