taeto

Quantum321 seems to prefer the hyphen. I have not seen it used by any others. Doesn't seem important either.

I just noticed this thread, so my answer unfortunately comes a little late to be of any use in the then ongoing tournament. But it surely will be implemented with success in future championships. This is an idea that I have had for years, actually I have been a proponent of this idea even before the time when, for several independent reasons, all my friends with whom I used to discuss the game mysteriously chose to severe our bonds, after which I have never heard from any of them again. The ultimate aim is to increase the number of scorings in the game, to ensure that enough entertainment is provided, which boosts the interest of the punters and provides financial benefits to the investors and advertisers. I will take a small percentage, if it is not too much to ask. I explain how to dramatically increase the number of goals scored in any game. And actually, in complete contrast to the current state of the game, the number of goals will increase the most in the top level games relative to the lower levels. If you ever attended games between teams consisting of pre-school children, you are aware that scores like 8-5 or 14-0 are not that unusual, and yet, noone would want to pay to watch those games on a huge stadium or televised. We will change this, so that results like that will eventually count as very low score between the top professional teams. Concerning the rules of the game on the pitch, make the following changes. None, no changes to the offside rules, how long the goalie can hold on to the ball, what happens when you slaughter the opponent, when you take a dive or remove the clothing on your upper body (special provision may apply to women's games), the size of the field or the goals, the colour of the grass. No changes whatsoever. Keep it exactly as it is. The single change concerns the point system. Nowadays the team that scores more goals than the opponent team gets awarded 3 points, in which case the losing team gets 0 points, with the proviso that if both teams score the same number of goals, each is awarded 1 point. At the end of a championship/tournament the points of all teams are tallied up, and the team with the most point gets awarded first place. In the case of a tie, there are tiebreaker rules that include the goal differences and the number of goals that the teams have scored during the season. In future, the team that scores more goals than the opponent team gets awarded 1 point, the losing team gets 0 points, and if both teams score the same number of goals, each is awarded 3 points instead. Again, at the end of a championship, the team with the most points, and in case of tie having the best tiebreakers, become champions. Imagine how much this simple rule change will liven up the games. Obviously each team is hugely motivated towards achieving a tied game to gain the maximum 3 points towards the championship tally. Doubtless almost every game in a championship with teams of professional players should end in a tie, and quite possibly all of them. With practically all teams having gained maximal points at the end of season and a level goal difference, the decision about first place comes down to the number of goals scored. In fact, in every game the strategy of both teams will have to be to score exactly as many goals as the opponent, and, subject to that, as many goals as possible. Final game scores of 50 - 50 should easily become the norm, if not more, so that will be more than one goal every minute. If that is not good enough for you, go and watch basketball instead. Surely as to NFL you can forget about their laughable score lines. Nowadays, the goalies are pesky creatures, as their objective is to prevent goals, which is counter to our desire. This will no longer happen. It is all in the best interest of both teams on the pitch that as many goals get scored as possible, so long as the score remains level. In fact it will become the primary task of a goaltender to make sure that the ball gets hammered into the net behind him or her as quickly as possible, before the game starts again with a replacement ball from the center spot. If the game did not have a bright future before, I feel certain that it has one now.

In the context of the question the above is maybe not a fair way to answer. In ordinary calculus you consider finitely many, say n, small sections of pencil, and you add up their lengths and get a length measure L(n) that depends on n. The integration means that you calculate a limit of L(n) as n approaches infinity, in a mathematically well defined and fully self-consistent manner. But there are not infinitely many small sections, only the fixed number n of them. The number n really is fixed and obviously finite, hence so is L(n). Calculating the limit only depends on the behavior of the function L that you consider. Mathematically it is possible to have an infinitely long pencil with a finite volume. Maybe it stretches from x = 0 and all the way up through x > 0 so that the volume of the segment of the pencil between x and x+1 is equal to \( 1/ x^2 \).

What happened to the math typesetting in that strange post though? Whatever it was, it is something I would want to avoid

Proof of a cardinal infinity sounds interesting. Will it be preceded by a few paragraphs that explain what such a thing is?

I feel that I should say something here. But I do not know exactly what. The Collatz conjecture is extremely difficult and hard to prove. Some people would say that it might be impossible. A proof or disproof of the conjecture would be extremely interesting. If you have an idea for a proof, you should certainly discuss it with somebody, and find out whether there is a hole in your proof suggestion. If you cannot find a mistake, it is still not a disaster if you send a written up version to a journal and it turns out to be wrong. Of course if it turns out to be correct it is just great, but I think you are not worrying much about that case anyway. The problem is that if you submit a manuscript to a journal, and it is not written in formally coherent way, then it could be summarily rejected, especially if you are dealing with a high-end journal. Which you naturally should be, if you propose to solve this conjecture. Is there some way that you can give more information about your work, without revealing enough to enable somebody to steal it? If you have a finished manuscript, I can recommend to submit it to the Electronic Journal of Combinatorics, of which I am an editor: www.combinatorics.org - we will take a look at it and recommend what you should do with it further, possibly guiding you all the way to final publication.

There are thousands of species of mayflies. I lived in Canada for a few years and they are very numerous there. I believe one nickname that they have in the US is "Canadian soldier".

Looks similar to en.wikipedia.org/wiki/Ephemeridae

I do not understand your comments. Yes, I am aware that a single example is only one particular case. However the example shows that even if given an uncountable set of sets each of which is also uncountable, then it may be quite easy to point to a choice set. You appeared to suggest that this seems impossible, which appears to be a comment about the general case. I tried to point out that those cases when you must use AC are special cases, and not the general rule. I probably misunderstood your original comment. What I think you could have said is that to a naive person it might seem that you can always find a choice set, whereas to a sophisticated person it is clear that no choice set has to exist unless something strong enough like AC can be applied. Uncountability does not enter into it that strongly. As another example, assume that a relation ~ is defined on the set of real numbers \(R\) so that \( x \sim y \) holds whenever the difference \( y - x \) is a rational number, i.e. \( y-x \in Q. \) It is easy to check that ~ is an equivalence relation, which means that \( R \) is partitioned into equivalence classes, each of the form \( [r] := Q + r, \) for some \( r \) that is irrational or zero. Every equivalence class is just a shifted copy of \( Q, \) so in particular it is a countable set of real numbers. Even though each set \( [r] \) is countable, it is impossible to prove the existence of a choice set for the equivalence classes without using AC. (A particular such choice set is called a Vitali Set and is important in measure theory.)

That does not sound right. You can never perform an infinite amount of operations anyway. But it is quite possible to get a choice set from an uncountable set of uncountable sets, by just a single "operation". Example: the Euclidean plane \( R^2 \) is the union of disjoint uncountable sets \( R(x) := \{ (x,y) : y \in R \} \). The graph of \( y = x \) is the uncountable choice set \( \{ (x,x) : x \in R \} \) of the sets \( R(x), \)which you get by application of a single axiom in ZF, no choice axiom needed. It is only sometimes that you can only get the choice set by using AC.

Your are right of course. However, the classical finite example considers the, presumably finite, collection of pairs of socks in your closet. Assuming that any two pairs are different, say, they have different colors, but the two socks from each pair look identical. You ask a friend, for whatever reason, to go and pick one sock from each pair. You cannot really explain to your friend which one to pick out of each pair; to say "take the left one" doesn't work as they are identical. If your friend is compliant and the closet not a jumbled up mess, they will be able to pick just one sock from each pair, and in the end you have a well-defined collection of different looking socks. The axiom of socks, sorry, choice, acts similarly, by allowing you to pick a set similarly, without any need to explain how it is done by any rule of picking.

Linear and semidefinite programming are fairly obvious examples.

You do not need any formulas. How about you think about what you are asked: how many times do you have to multiply .5 = 1/2 to itself to get 1/32? How would that work? Like 1/2 x 1/2 = 1/4 does not quite get there, neither does 1/2 x 1/2 x 1/2 = 1/8. You need to multiply a 2 together with itself enough times to get 32 as the answer.

I thought that simultaneity is taken into account by specifying that the measurement is taken "now", at time t=0. Which is when you look at your watch right now, and when the boundary of the observable universe is 46B LYs distant. The integral is taken over all points in spacetime that have the same value t=0. Yes, at that time you can measure density as what it is locally ( which would seem sensible), or what it looks like from our frame. Density and volume could be more of an issue, since volume decreases with distance, density creeps up. Possibly the density integral diverges.

So this is not trivial because "density" depends on the frame. We observe an amount of matter in dilated space appearing very dense if it moves away fast? Then there could be two different measures applied to density. A "relative" measure, where we calculate the amount of matter per unit of space in our frame. Or an "absolute" measure in which we calculate the amount of matter per space in the local frame that moves with it. If we know the speed of each receding point, and assuming that the local density is known, then it should not seem that hard. For each point in space, we get a density either way, and we can integrate?

This confuses me too. It is said that the universe is 13.8B years old since BB, and the radius of the observable part is about, what, 46B light years now. So it is right to say that if you assign time t=0 to our present location, then you assign t=0 to points at the border of the observable universe as well, by saying that they have moved there "by now"? Why is it then not simply a matter of integrating density over space at time t=0 to figure out the center of gravity of the observable universe as it is "now"? What am I missing?

I agree that this is what should be the idea, physically. There is still room for confusion, because the questions whether "space" or "spacetime" is finite or infinite appear ambiguous. Surely space, as a mathematical abstract that serves to keep track of the coordinates of things that are happening, has continuum size, regardless of geometry. That is the way it is used in theoretical physics. No? You would probably explain that "the universe" consists of both "the space(time)" and the stuff that occupies it, but space itself is just one single thing, as opposed to the entire totality of its points? After all, space is there in the universe, so it should count in some way to the totality of things. When it is explained that a condensate has the property that adding or removing a single particle does not change it in any way, it is clear that this means an infinite amount of single particles have to be present. But I have to perceive this statement as just an approximation, to say that it is close enough to what actually occurs. It seems to me that he never explains this latter part, but, after all, he mostly just lectures to old people that are not necessarily keen to actively force the distinction. So yes, the exact definition that he usually gives definitely does imply the infinite amount. Whether confined to a finite part of space is however not completely clear; because of SR I would guess that finiteness is implicitly implied. I can come up with spontaneous particle-antiparticle formation and annihilation as a mechanism to quickly create lots of stuff in a small space. Is there any known bound to how many pairs may be created within given space and time? Are there more reasonable/effective candidates?

I know the topic is old hat. I have read through the thread, and notice that most answers do not speak of this question in the same spirit as I naturally tend to think of it. That is, coming from the mathematics side. Mostly it is perceived as a question of size, and that is fair enough. If the volume of the space part of the universe is infinite, then it is reasonable to consider it to be infinite. Even if that is not the case, then it is still possible that the volume of the entire spacetime is infinite. Suppose though, that the volume of space is finite, and maybe even of spacetime as well. Could not the contents of space still be an infinite amount? In particular, is the amount of physical content within a 1x1x1 cube (resp. 1x1x1x1 spacetime cube) necessarily finite? I have to add some motivation to that question. First, in mathematics, I am absolutely happy with the fact that in Euclidean space, the modest straight line segment between 0 and 1 contains an infinite amount of single points, even an infinite amount of rational valued points. And if you consider all the Euclidean points in this segment, there is the same amount of them as in the entire Euclidean space \(R³\). It seems to me that space itself easily contains an infinite amount of things, namely of single points. After all, even physicists keep referring to a geometry of space and spacetime that necessarily consists of a continuum of points. Some physicists that I met, though I am not sure if they are just pop-sci types, explain that even in the interval between 0 and 1 there is only a finite number of Planck lengths available, hence the interval itself should count as finite. To me that sounds pretty simplistic, as it seems presupposed that you can determine the beginning and end of each interval of Planck length in which something uniformly happens throughout. Whereas you can never hope to figure out what happens on a smaller scale. This is quite reasonably true in QM, but it is not one of those things that you confirm experimentally, and it seems it could be changed by a quantum gravity theory that extends current QM. Second, when I listen to Susskind explaining BHs, he explains the concept of a condensate so that it has the property to contain an infinite number of identical particles. And this concept is used to explain how QM works in some situations. Is there any reason that has to be b****cks? It seems fine if the particles are massless with a momentum that adds up to near zero, or not? And if the universe contains an infinite amount of individual particles, then it also has to count as being infinite?

If you did it and got ~ 3195.323, then it looks like you did it correctly. Or are you just referencing the answer known from a solution book?

If you also want to see an \( e \) you can have \[ e^{\pi i/5} = \frac{\varphi + i\sqrt{3-\varphi } }{2}. \] It is cheating, since both sides are just a \( \sqrt[5]{-1}. \)

Is the Ramsey quote really correct, did Ramsey write anything that stupid? A "tautology" is usually a statement that is always true. If you build a theory axiomatically, then the axioms are always true, by construction, in that theory. And anyway, supposing you throw out a number of axioms, it of course would make arithmetic statements even more undecidable than they are already. Eventually you might no longer be able to prove that this is so. But if that is your intention, then you could instead also close your eyes, put your fingers in your ears and go LALALALALA and avoid being disturbed by facts that way.

But it is an actual boundary more than just an interface. A (non-rotating) black hole at a given time has a precise Schwarzschild radius that depends on its mass, and the event horizon surrounds the singularity spherically at that precise distance. I don't think you are supposed to imagine anything being fuzzy. In which case it qualifies as a 2D surface.

He means arctan (times a suitable constant). It is different from a logarithm in an important respect. Although both functions will keep increasing, the log is unbounded, whereas arctan will approach an upper limit. So it depends on which of those properties you are aiming for.

Do you consider the event horizon of a black hole to be a thing that exists in the real world? It might fit the bill pretty close, no?

That is actually a good question. I have a firm memory that tells me that it at least used to be a very explicit requirement that every proof can be verified within the theory ZFC. The rules were modified in September of 2012, however. I would be very interested to know whether the requirement got changed, and why. It would clearly be absurd to allow a proof in a theory of your own choice, as you might simply add as a new axiom the exact statement that you desire to proof, or at least some more disguised statement which would imply it. But I definitely surprised by your observation, so I salute you for that. Anyway, the official description of the problem is written by Stephen Cook and it does define a Turing Machine in the language of ZFC. That makes it unlikely that a proof outside ZFC is in any way meaningful. But I am open to suggestion. Do you have a theory to propose different from ZFC in which you would suggest to attempt a proof? Scott Aaronson has a discussion of the dependence of the NP=P question on ZFC on page 26-28 of www.scottaaronson.com/papers/pnp.pdf. If you submit anything rubbish, they would obviously turn it down. If you submit an actual solution they would quite obviously be delighted, no matter what degrees you do or do not have, heck, it would not matter whether you finished high school or not. Did you ever submit a paper to a journal, or are you just guessing here? Not sure what you mean by that. You quote an informal description of a DTM that requires the read head to move either backwards or forwards every time it needs to read from a different cell. I also do not know what you mean by saying that a problem "requires an infinite tape". Either example of a TM requires an infinite tape, otherwise it is simply not even possible to accept all legal inputs. If your TM factorizes integers, and its tape only has room for inputs of N bits that are available on your tape, then what will it do with an integer that has to be written with N+1 bits?