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This was in response to your question about what happens if we continually halve the distance to the end of [0, 1). But if you accept that, why are you still confused about halving the distance between p and q? In set theory, the continuum is just another name for the set of real numbers. In topology, a continuum is "a nonempty compact connected metric space." Either way, it's not entirely helpful to try to reason from the everyday or philosophical meaning of the word. https://en.wikipedia.org/wiki/Continuum_(topology) By the way, the set of real numbers is not compact. So according to Wikipedia, the real numbers are not a continuum. That's contrary to pretty much everybody. You have to take Wikipedia with a large grain of sodium chloride. In some ways yes, in some ways no. In terms of cardinality, they are exactly the same. In terms of length, they are exactly the same. In terms of topology, [0,1] is a compact set, which has many important properties that [0,1) lacks. For example any continuous function on a compact set must necessarily attain its maximum and minimum. This is not true of [0,1). So when you remove the end point some things don't change and other things do. Topologically, removing that one endpoint makes a huge difference. https://en.wikipedia.org/wiki/Compact_space Is this a physical onion, or an imaginary onion in your mind? If it's a physical onion, it does not have infinitely many layers. If it's imaginary, I don't know what you're imagining. But explain me this. Did you understand my earlier demonstration that no matter what number you claim is the largest in [0,1), it turns out that it is NOT the largest. Did you understand that? If so, why are you still tossing out imaginary onions? And if not, which part is unclear or unconvincing? One perfectly sensible response on your part would be, "Oh, I see. There can not logically be any largest element of [0,1). I shall adjust my intuitions accordingly." That's the purpose of the exercise, to sharpen and correct our intuitions. Not to talk about hypothetical imaginary onions after you've been shown a proof. If you dispute the proof, let's focus on that. Having seen the proof, why are you still insisting on an intuition that is falsified by the proof? Let's nail down the understanding of the proof that there is no largest number in [0,1). Once we do that, then it will be clear that all intuitions to the contrary are inaccurate. What he meant is that in higher set theory, we can study transfinite numbers, like the transfinite cardinals [math]\aleph_0, \aleph_1[/math], etc., and the transfinite ordinals ω,ω+1, , and so forth. These are far outside of the scope of our discussion, but that's what your professor was referring to. That was @Genady's comment on page one of this thread, credit where due. But if you understand that we can continually split the difference 1/2, 3/4, 7/8, etc., why are you still unclear about this? It's the same idea. Well FWIW nonstandard analysis is also a part of ZFC. But we are talking about the standard reals and need not go any further than that.

I can’t answer this, as evolutionary biology isn’t my area of expertise. It’s a difficult subject also because the autism spectrum is so broad - there are some like myself, with very few to no situational support needs, and then there’s a sliding scale of increasing severity right up to forms of autism that make independent living (never even mind independent survival) practically impossible. So it’s hard to generalise. I’m speaking only for myself now, and perhaps those with similar profiles and predispositions as me. I think there might be an evolutionary advantage precisely because people like me don’t fit into the mainstream. For example, my sense of purpose, meaning, and well-being is not contingent upon social acceptance and belonging - things like how many friends I have, social gatherings and occasions, belonging to a certain group (or not), being around other people etc etc are simply of very little importance to me. This might at first glance sound like an evolutionary disadvantage, but think about it - it frees up enormous amounts of time and energy that can then be re-invested into other pursuits. I don’t know if there are statistics about this, but I bet that, among people who have made important contributions in their fields - the arts, sciences, literature etc etc - a disproportionally large amount might be found to be on the spectrum, or at least have autism-like traits of some sort or another. This is because such people are more likely to engage deeply in pursuits not directly concerned with survival and procreation (which is what social preoccupations are ultimately geared towards). I think society benefits from this kind of archetype - the ones who can stand on the sidelines, look back onto the mainstream from a more neutral and wider external perspective, and pursue “higher” things and unusual ways of thinking. I think there’s an evolutionary advantage for the group as a whole in having such individuals, because they function like a mirror that reflects back the forest when all you yourself are usually able to see is the trees, due to your own day-to-day involvement. Such individuals are often simultaneously despised (because they don’t fit in), and valued (for their contributions, often only posthumously), and sometimes burned at the stake; but whatever the case may be, their perspective is an important one. These are just some of my own thoughts, I’m making no claim to any academic truths here. Yes, but it’s not just that - it’s a theory of the world, including the physics side of things. When we are building models in physics, then these are necessarily models of aspects of how the world appears to us. They are thus models of aspects of another model, namely the reality-model that our minds create for us. We all tend to agree on certain aspects of that generated reality simply because we all share a similar sensory apparatus (plus its extensions), and a roughly similar neurophysiological brain structure - thus the boundary conditions are similar, meaning the resulting reality-model is also roughly similar. The reality-model of an organism that evolved under sufficiently different boundary conditions may potentially be quite different from ours - an example from sci-fi literature that comes to mind are the heptapods in Ted Chiang’s “Story of Your Life”, whose minds do not employ the principle of temporal sequencing in constructing their reality-model. I know it’s just a story, but it’s an interesting example. So what happens if the boundary conditions vary? I wrote about autism and social “mind-blindness” above - so what is actual reality here? Are social relations and intuitions real, irreducible aspects of the world - or are these contingent add-ons that your neurotypical brain artificially generates, and it is actually my own mind-blind autistic self who sees things as they really are? Or how about this - in addition to being autistic I am also a synesthete. Words to me have colour, texture, size, spatial orientation, and sometimes temporal extension. These, to me, are not associations (e.g. sky=blue), but intrinsic properties of the words themselves (so for me sky=off-white, smooth and cold like marble, angled backwards and to the right), like spin and charge for an elementary particle. For me this is so intrinsically normal that I am pretty much unable to imagine what experience would be like without these attributes - I only know intellectually that most people can’t experience this the way I do. So who perceives “actual” reality here - is the concept “sky” really smooth and cold, and you are all just blind to that? Or does my brain adds this on randomly? Who’s right and who’s wrong? Or is the entire concept of “reality” just a constructed idea, the meaning of which is strictly contextual? Now think about the wider implications for physics - it makes models of a model. But how do we know, within how the world appears to us, what is an actual part of exterior reality, and what is an add-on by our brain? How can we distinguish, in the absence of having an external reference in the form of other reality-modellers against whom we can compare our reality? Do (e.g.) time and space really exist in the way we experience them, or are they just convenient representations to impose order onto a set of data, like the windows on the GUI of your computer? Are there other ways to structure that same information? Or are there aspects of exterior reality that are not being represented in our model at all, not even by deduction or induction, perhaps because they are irrelevant to our continued evolution? Does the way we do science thus say more about ourselves and how or brains make reality appear to us, than exterior reality? I think these are important questions to consider not just in philosophy, but also in the foundations of science - just focussing on the model, while ignoring what the model is actually about, and who constructs it, might be misguided and eventually come back to haunt us. I don’t feel this is spoken about enough in the physics community, or even taken seriously.

You are standing on the ground holding a bowling ball, and it weighs about 12 to 16 lbs. It is a strain to hold it at arms length. Now go parachuting while carrying that bowling ball; you will find it weighs nothing after you jump out of the airplane. Yet you and the bowling ball are still falling due to gravity. What happened to your force, Martillo ???

IOW you observe behavior. The force of the air is inferred, but how do you know that’s the reality, and there’s not something else involved? How can you be sure it’s not invisible fairies moving the leaves?

Cue the spam link as next post from another newly created account…

Or this one: PS. Took me 30 s.

I don't see how "dry air and nose pickling shouldn't be the cause of nosebleeds" is suggested by "none of them said that they had nosebleeds". I can only see how the latter suggests that the nosebleed condition is not very common.

Try this one: Japanese brain aging test. Use the stopwatch.

By "fundamental" I mean resting on minimal/general assumptions, although I know how slippery that concept can be. Yes, theory of the mind --understanding that others have minds probably like we feel we ourselves have minds, and acting accordingly-- is one of the most important adaptive pressures that shaped the evolution of the human brain. I don't have the sources at hand to assert this, by I know from the reasoned arguments of many scientists of human evolution I've sampled through the years. I am in no doubt that there must be a reason why Nature has kept the genes that give us autism, which at first glance looks like just a cognitive impairment. Look closer and more can be seen. From my experience, from what I know from you, and others like you: Genuinely caring individuals, who suffer when they see conflict, generally devoid of manipulative intentions. Never foul players, sincerely concerned about problems, both human and technical, their own, and those of others. Hard workers, obsessive in a way that can be very productive, given the proper outlook, focus, and advise about how letting go when the time comes. Sometimes we're discussing something and we're being petty and narrow-minded. And here comes Markus Hanke and shines his light. All the pettiness is dissolved. The problem is re-focused to what the problem is. I guess that's why you lot are here for. Don't look now, but you activate us in a direction that --always in my experience, mind you-- usually is a good one.

I wonder if those studies include the impact of capitalism on the elderly. In the US, we spend our whole lives saving for retirement, paying into Social Security, only to find several well-oiled, privately owned industries are just waiting to sink their hooks into your savings. Medicare is rife with all these private schemes that suck the effectiveness out of your retirement savings (I get invited to steak dinners so they can sell me on my government's medical coverage). Even if you can avoid scammers, there are too many legal ways to fleece the elderly.

To a very good approximation, the effect of temperature on the density of air and helium is the same (in percentage terms). The balloon floats because its average density (including the walls of the balloon)is less than that of the air. As the balloon rises the air gets less dense. There comes a point where the density of the balloon and the surrounding air are the same and, at that level, it stops rising. It's made slightly more complicated by the fact that the balloon changes volume (and thus density) as it rises.| With mylar balloons, once the envelope is "full", the volume stays pretty much constant (until they burst). High altitude balloons are launched looking rather empty, to allow for expansion as they rise. https://www.forbes.com/sites/elizabethhowell1/2019/07/05/google-affiliated-company-sends-high-altitude-balloon-aloft-for-223-days/?sh=484528407edf

The ekpyrotic model proposes that the universe has no beginning or end. Instead of a bang, what happened is thought of as a "Big Bounce", the moment at which the universe, which was slowly contracting to an incredibly, but not infinitely, small point, bounced to expansion. There is no beginning of time. Instead, our visible universe exists on one of two four-dimensional "branes" floating in a five-dimensional space.The ekpyrotic universe theory asserts that the cosmos exists as a product of an endless procession of creation, destruction and recreation. This model theorizes that the universe has been expanding and contracting repeatedly over time scales that make the 13.7 billion years that have passed since the Big Bang seem like a second.

There are at least 10 sextillion stars in the entire universe. That's 10 followed by 22 zeros. This number is thousands of times larger than all the grains of sand on all the beaches on Earth.

Light is made out of small quantum objects called photons. When you turn on a lamp, the light bulb begins creating and emitting trillions upon trillions of photons. Photons are in a class of quantum particles known as bosons.

As I move (spin) into the unknown (spacetime fabric) I encounter disturbances(quantum fluctuations) I stretch my primordial arms(primordial Higgs like field) to swim, you won't catch me, though the arms drag (mass) me, they are still primordial , I become a ghost (NEUTRINO),love me like that am a wanderer.. neutrino oscillation.pdf