joigus

I think that's because \( 1-1=0 \).

Rather: \[ \textrm{int}C\neq\textrm{Ø}\Rightarrow\textrm{if }x\in C\Rightarrow B_{\epsilon}\left(x\right)\subseteq C\Rightarrow\mu\left(C\right)\geq\mu\left(B_{\epsilon}\left(x\right)\right)>0 \]

This is the part I do not understand: And this is the part I do understand:

Deterring doesn't work 100% of the time. But it does in many cases. We need @Eise here. Wittgenstein or not.

Is it Lebesgue day today?

For the first one I would try the inverse direction; something like (as @mathematicsuggests), intC≠Ø⇒μ(C)≥0 Think open balls. I see you already have good help, so I'll leave it at that. Cheers.

Testing more stuff. \[ f\left(x\right)=\begin{cases} 1, & x\in\mathbb{Q}\\ -1, & x\in\mathbb{R}-\mathbb{Q} \end{cases} \] Testing more stuff. \[ \textrm{int}C\neq\textrm{Ø}\Rightarrow\textrm{if }x\in C\Rightarrow\in B_{\epsilon}\left(x\right)\subseteq C\Rightarrow\mu\left(C\right)\geq\mu\left(B_{\epsilon}\left(x\right)\right)>0 \]

Not hours; minutes. But many. Why? For all crackpots out there: https://www.gapingvoidart.com/

I think you have a point, @Sensei. I think what you're saying is "it's safer to interpolate a curve with three points than interpolating it with 2". Nobody would expect the results with anti-muons to be any different, though, because the standard model is CPT invariant. If anti-muons gave a disparate result... Now that would be a surprise! What I don't know is how difficult the experiments with taus would be. I'm guessing a lot more difficult, due to masses and lifetimes.

I'm more inclined to assess this as incontrovertible evidence of new physics beyond the standard model. But I'm reluctant to salute it as incontrovertible evidence of a "fifth force" just yet. For a fifth force to be there beyond any doubt, there would have to be evidence of new decay modes revealing brand-new gauge bosons, with new quantum numbers. But it is true that it's very difficult to conceive of a different gyromagnetic ratio of higher-generation leptons without anything dynamical being involved. The calculation of g-2 involves radiative corrections, essentially sums on all the gauge bosons "virtually flying around", and it's a dimensionless factor. If the gauge bosons are the same for different families, I see no reason why the gyromagnetic ratio should differ unless there are new radiative modes involved.

Yes, but that's not the point about matter-antimatter asymmetry. Call them what we may, the thing is there are considerably more electrons than their counterparts, and protons that their counterparts, and so on. There is an unbalance to one side, so to speak.

Beautifully put.

You need a mechanism that smooths out the universe to the presently known value, and does it at superluminal rates without violating causality in a local way. The way to do that is an expansion factor in cosmology consistent with GR. That's what inflationary models do. The fact that monopoles are swept out of sight is a bonus of inflationary models, rather than a robust argument, I think. The point being immediate generalisations of the standard model of particles physics (grand unification theories, aka GUTs) do predict these very heavy particles. So we can still pursue them (GUTs) while contemplating an explanation of why they (monopoles) aren't anywhere to be found. You need to study what the present models do in order to propose a wannabe cosmological model with any chance of being seriously considered. I hope that was helpful. I don't think a BEC will do the job, honest.

The rationale for the current inflationary model --as envisioned by Alan Guth, and developed by others--, is the need to explain certain observational facts: 1) The universe is large-scale homogeneous (this conflicts with causality: how did causally-separated regions equilibrate?) 2) The universe is extremely flat (why is it so stretched-out?) 3) Absence of heavy particles predicted by GUTs (monopoles) Your model should address these questions.

Apparently there will be a period of radio silence...

That's like saying that insects are many times stronger than ants. Humans are primates.

Indeed. That's in the other half of what I intended, really. They are no saints; they are no villains. They're neither.

Typically-prey animals are no saints. A male buffalo in heat will have no problem trampling to death a calf of his own species. Male herbivores sometimes kill each other when they're fighting for females. Many herbivores are highly territorial, and become very aggressive towards anything trespassing what they consider their territory. What's this nonsense about animal ethics? Let alone good (prey) vs evil (predators)... Rodents are known to eat their young in times of environmental stress. Animals in the wild and under high pressure to survive cannot afford the least last protein to go to waste. Nature can turn a cuddly rabbit into a frenzied cannibal eating its own kind. And back to OP. Cannibalism has never been a common practice among modern humans, for all I know. When it's been practiced, it's been mainly due to famines, or to ritualistic behaviour in ancient societies. The picture of a human looking at another and thinking "mmmm yummy!" is a caricature of a much more elaborate, complex, and relatively rare phenomenon.

Loved your video, but here's where we disagree. I think there are mathematical patterns even in the cultural world. Wherever or whenever we don't understand them, I think it's because the pattern has not been discerned as yet. In that sense, I'm Platonic perhaps. I think mathematics underlies everything.

Sorry, what are those? Very interesting. Thank you. Concerning the reason why we're so "tuned" to being pleased by sequences (or longer overlaps, as in chords) of frequencies that are related to one another by integer numbers; my guess is as good as anybody's. But I don't find it very surprising. What I find even more surprising is the fact that there seems to be this fixed reference of a central note. Anybody who's attended a classical music concert --or Renaissance music--, which I do quite often, is familiar with the protocol of all the musicians tuning their instruments to A major when the concert is about to start. What defines A major? Even more amazing, apparently there are people who have absolute pitch. I know about this because I have a friend who is a physicist and advanced piano player who has it. These gifted people can tell A major with no "external reference", so to speak.

An image is worth a thousand words:

There's an even more terrifying possibility... I'm sure it hadn't even crossed your mind.

Just for the benefit of other users who are presumably going to waste a lot of effort here. I already explained what makes musical notes special --a reasonably centrally placed A major, which is quite audible for a large range of people, but possibly arbitrary as to its exact value--, and then the harmonics, which are defined as integer multiples or fractions of it. Also gave hints that the question is very old --goes back to the Pythagorean school--. I equally argued that light is very different, because we don't intuitively perceive it as "frequencies of something oscillating", although it is, in the last analysis. Also hinted to the fact that when you play a note after another, they overlap, and you notice that they are in sync. All these points went unnoticed. The OP doesn't seem to care one way or another. In fact, this thread could well end up being about gravity. Who knows.

You're confusing levels of explanation. Light (photons) makes up what we call "to see." A very complex large-scale phenomenon. You don't "see" a photon (light). Photons excite receptors, and nerve impulses project those excitations in your visual cortex. That is what we call "to see." One single photon most of the time probably is not enough to excite a photo receptor. Sometimes we even see light where there is none: https://en.wikipedia.org/wiki/Phosphene Already the Pythagoreans noticed that humans find sequences of notes related by integer multiples specially pleasing. Why is that I'm not sure. I'm not even sure if anybody knows. But I think it's probably related to the fact that sounds are very low-frequency in relation to light. So you don't have a chance to time-resolve different kinds of light. Let me tell you what I mean. Heisenberg's principle (actually, a general theorem of Fourier analysis of waves) can be illustrated with a piano. It is well known that to make the pitch of a note more definite in you mind, it is necessary to make it last longer the lower the frequency is: \[\triangle\omega\triangle T\sim1\] Here \( \omega \) is the frequency bandwidth (in cycles per second), and \( T \) is a characteristic time that represents how long you have to let it last to make it distinct (a sort of "timewidth"). When notes sound one after the other, for a split second you hear them as a chord (before they die out) during the time they overlap in your hearing. Because sounds have much much lower time frequencies, your brain has a lot more of a shot to tell that something is "in tune" there. Not so with light. With light it's energy levels of molecules that have to do that job. You don't have this perception that something is oscillating more slowly or more quickly.

No: Reading what I said could be helpful.