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I'll take a run at this. I think the key issue is that there is no completeness theorem for second-order logic. In general the OP's post doesn't mention the key distinctions between first and second order logic. OP should give this a read, perhaps there is some insight to be had. https://plato.stanford.edu/entries/logic-higher-order/ In particular, note section 5.2: "We shall now see that there is no hope of a Completeness Theorem for second-order logic." Ok, that's a funny way of putting it but I know what you mean. Ok. But you are conflating syntax (axiom systems) and semantics (models). That is, the 180 degree theorem follows syntactically from the axioms of Euclidean geometry. That's a purely syntactic fact. And it's also the case that the theorem is true in any model of those axioms. That's a syntactic fact. Two different things. In first-order logic, a theorem is provable (syntax) if and only if it's true in every model of the axioms (semantics). This is Gödel's completeness theorem. But beware, as the SEP article I linked above indicates, there is no completeness theorem for second-order logic. I confess to not knowing much about the fine points of second-order logic, but I suspect the OP's questions relate to this fact, that second order logic does not have a completeness theorem. The terminology is that an axiom system with a unique model up to isomorphism is categorical. And one that has non-isomorphic models is non-categorical. OP, please read this. https://en.wikipedia.org/wiki/Categorical_theory Now the first-order theory of Peano arithmetic most definitely does have nonstandard models. For example the hyperintegers of the hyperreal numbers are a nonstandard model of PA. They include the usual finite natural numbers 0, 1, 2, 3, ... as well as all the infinite ones. However the second-order theory of PA is categorical, and perhaps that's what you are referring to as Dedekind's theorem. So this is an example of where you need to distinguish between first and second order theories. There ARE non-categorical models of first-order PA; but none of second-order PA. This I believe is the crux of your concern, if I'm understanding your post. Yes, but note that the real numbers axiomatized as an infinite, complete ordered field is categorical. That's because completeness is a second-order property. It quantifies over subsets of the real numbers, not just individuals. That is, the least upper bound property says that all nonempty subsets of the reals that are bounded above have a least upper bound. Since we have quantified over subsets, that's a second-order property, and then it turns out that the second-order reals are categorical. This is where my knowledge ends. Since there's no completeness theorem for second-order logic, there must be some axiom system in which something is a theorem that's not true in all models, or true in all models but not a theorem. I don't know any specific examples or anything more about it. EDIT -- this isn't right, see below You're asking if there are undecidable statements in the categorical theory of second-order PA. This I do not know. A related question of interest is whether there are "natural" statements of arithematic that are undecidable (presumably in first-order PA). Harvey Friedman has been searching for examples of such. Here's an overview. https://plus.maths.org/content/picking-holes-mathematics Well that's what I know about it, hope something in here was helpful. The key is that first-order theories are never categorical and second-order theories sometimes are. But there's no completeness theorem for second-order theories, leading to all the aspects of this that I don't know, and to the question you're asking. ps -- Aha, a clue. I was perusing the Wiki article on categorical theories at https://en.wikipedia.org/wiki/Categorical_theory and it says: Every categorical theory is complete. However, the converse does not hold So this tells us that IF a theory is categorical (has only one model up to isomorphism) THEN every theorem provable from the axioms is true in every model of those axioms. But if the converse fails, then there is an axiomatic system with a statement that is true in all models of the axioms, but that is not provable from the axioms. Well, "today I learned!" pps -- The Wiki footnote leads to Carl Mummert's answer to this math.SE question: https://math.stackexchange.com/questions/933466/difference-between-completeness-and-categoricity/933632#933632 Mummert gives a concrete example of a comple, noncategorical theory. This doesn't seem to be the same thing as what we are looking for, a categorical, non-complete theory. There may well be clues on this page though. Reading a little more of Mummert's response, he's using "complete" in a different sense, that every statement is either provable or its negation is. That's not the same kind of completeness as in Gödel's completeness theorem, which says that a statement is provable if and only if it's true in every model. These two subtly different meanings of complete are yet another confusing aspect of all of this. So I was wrong about what it means for a theory to be categorical but not complete. Nevermind that part. These are deep waters and I got in a little over my head. From the Wiki article on completeness: https://en.wikipedia.org/wiki/Complete_theory

Stellar aberration’s basic explanation was known since ~1725, which precludes us being at rest with respect to the aether. We had to be moving, but the M-M experiment showed that to be wrong. https://en.wikipedia.org/wiki/Aberration_(astronomy)

there's Kahn academy, they walk you through problems step by step with videos and quizzes to test your understanding.

Hi bayukutten. I would have thought most maths textbooks for a school Mathematics course for people between about 11 and 15 years of age would have a section on signed numbers (often called positive and negative numbers). I haven't reviewed all of these books so I wouldn't like to single out one book as being better than others. You obviously have internet access. I find the BBC Bitesize educational resources are usually of a high standard and free to access. So I would recommend their site: This is aimed at British key stage 3 (age range 11 to 14): https://www.bbc.co.uk/bitesize/guides/z77xsbk/revision/1

Or you take the engineering approach and just read-off provided numbers: https://globalsolaratlas.info/map. If I remember correctly, the tool even has a "mark an area and integrate" functionality. I'd like to say something constructive here. But I find it hard to make out what this thread is about, or to add anything meaningful on this very vague level. I mean: Yes, solar panels generate electricity. Yes, you can put them on rooftops. And yes, there is lots of sun in the equatorial regions. And to Swansont's post: Yes, there are problems in detail. Transport and storage are somewhat generic problems, and they are at least easy to handle - any scenario calculation in the planning phase will implicitly include them. Rather specific problems to solar power in the Saharan regions seem to be sand, corruption and a perception of modern-day colonialism when rich white guys try to tell Africans what they should be doing. The idea of exploiting the solar power opportunities in the Sahara region is obviously not new. My personal favorite idea in "think big" is a world-grid with a solar power belt around the whole equator, btw. In Germany, the Desertec initiative was very well known. They planned to generate electric power in Africa and export it to Europe (sounding like modern-day colonialism: check). To my knowledge, the project died in 2014 when most major industry partners quit. I don't know why it failed, but the common rumors are about drop in renewable energy generation costs within Europe and worries about generating your power in regions that are considered politically unstable (-> Arab spring and the civil wars that followed and are still ongoing).

First, that is a very incomplete reading of how SARS-CoV-2 affects our bodies. Damages are not exclusively to cytokine storms, and there has been some discussions whether it is really relevant to the observed damages. There seems to be an association with severity but then there is also the question of the triggers. That being said, there are always interactions with the immune system in vivo but it is not the sole source of damage. What is known is that the virus attacks several organs, including the brain stem and at in vitro studies show that cellular damages also occur. Moreover, there have been immunomodulating therapies which in some cases improve outcome, but as it turns out, it only works for a fraction of the folks. Others still die under that treatment (or get secondary infections and die from that). And very obvious examples are immunocomprised folks that die from COVID-19. In fact, they can also suffer severe consequences from otherwise fairly harmlesss viruses, so no, making the immune system blind to a virus is generally not a great idea. In addition, the viruses that we carry and which do not make us sick anymore often have mutations that reduces their virulence- our immune system has not changed in that regard (and again, except for inactive ones, they are generally only harmless if your immune system still works). As such an one-sided view that viruses do not harm us only our own immune system is simply wrong or at best misleading. It is like saying electricity does not kill us, it is the heart failure. Or the fall is harmless, just the landing happens to be lethal. To be clear, SARS-CoV-2 can be lethal for many folks and let it roam through your body without an immune system to keep it in check is pretty certainly deadly. There is no evidence that rapidly proliferating viruses in your bloodstream, organs and nervous system are in fact harmless. Moreover, the immune system is easily one of the most complex regulatory system in our body. All mechanisms to modulate our immune system are blunt instrument, including vaccines. While many, many folks are working on it, precision modulation of the immune system is still science fiction. It is probably also the closest to a panacea that we could get, if we get it. As you mentioned, this is not the time for high-flying dreams. If folks are unable to the simple things to keep each other safe, a vaccine is really our only option. We (i.e. most countries and their population) failed to take it seriously and since this is not a movie, there won't be a sudden miracle cure delivered by Arnold Schwarzenegger. In fact, if we had taken it more seriously we would not have needed a vaccine. But obviously that was too much to ask. Folks are dying, and if folks would just keep their friggen distance instead of dreaming about sunshine, bleach or magic we would not be here. Edit: if that sounds angry, it is because I am. A good friend lost a parent. I am afraid for my parents and things go exactly as everyone said it would if we are complacent and mess up. And now some folks are surprised and some are protesting the need to do the absolute minimum to protect your neighbour and community. Plus there are students that do not care because they are effing young and do not care that they potentially kill folks around them.

It doesn't matter if you are an atheist or a believer. What really matters if you are a good *) person.. I know many good atheists, and many evil to the core, believers (claiming, pretending or fooling themselves, to be good).. *) The problem is that for different people "being good" means something else.. For instance, moral behaving like having regular sex without marriage, for religious extremists and fanatics, means being evil. At the same time they cut heads of disbelievers, murder people, force people to abandon their religion, impose their views on others, terrorize people, are attempting to stupidify people (e.g. "boko haram" means "(Western) education is forbidden"). etc. etc. For me, obviously, having sex without marriage is not existing problem (i.e. it is not a sin, or an evil act). Without sex, nobody would exist in this world, so it cannot be evil. For me, forcing somebody to abandon religion through terror, is evil act. Conversion must be voluntary. Otherwise it is worthless fake faith.

The atoms or if I use as an example the water molecules pretty much absorb light of all wavelength ranging from radio wave to gamma ray(perhaps), the important part is you want to lock the atom in place, like trying to stop a spinning globe in place, so if the wavelength is too long, like if I touch the globe once every second, it would not stop as oppose to using an x-ray(the shorter the wavelength the better), the thing is it needs to be applied in equal from six directions(if not 8), or else it would get push around in the opposite direction of the laser. So alignment is very important. If you decide to switch to incandescent/black body radiation to light up the atoms it could work, but you still need good lasers' alignment from all directions(so you need some type of filter to get a coherent light beam). Wish you luck

ARE WE GOING ABOUT COVID-19 THE WRONG WAY Should we not be developing a vaccine that tells our immune systems to ignore the virus, not fight it. After all the virus can't harm us, our own immune systems do that by cytokine storms and leaving infected part of the body to fight the virus that can't harm us any how. we are festooned with viruses that have took millions of years for us to ignore.