Col Not Colin

Hi all and thanks for these replies. I'm cautious. Trying to identify the motivation for online activities is key to staying safe and savvy. Volunteers sounds good but it must cost money to keep a server running. Asking where the funding comes from seems a sensible question. Good news, I'm not planning to complain. At the moment my main motivation is not wanting to be a part of any activity that shouts down newbies. Well, thanks Blake. You were probably trying to do a good thing. I think that's what I was looking for, a place for discussion about science. It's got to be friendly though, one of my sons is going to spill their soup everytime, if you know what I mean. OK. Probably fair. I'll exit now. <-- Late editing. That bit's ridiculous and melo-dramatic even by my standards. There's no implication that it's just because of iNow.

Hi everyone. Is there an "about" link for this forum, similar to the type of thing most websites provide? Can I ask the following sorts of questions: 1. In a short paragraph, what does this website or forum aim to do? 2. What affiliations or structures exist (e.g is this a subsidary of We-are-foums-dot-com)? Where does the funding come from? 3. Is there a registered office and/or a conventional postal address for this organisation? 4. Is there an internal complaints procedure? - - - - - End of introduction, setting the scene etc. Informal, chat type discussion to follow --- (Briefly) What is the history of development? Did one person start this forum as a blog, for example? Does one person still have over-all control? This will help to inform us what kind of thing is happening here now, what to expect and indeed whether we want to put in any time here. How have the existing rules developed? Is there scope to modify those rules according to user requests? If I was going tp pick one specific thing to discuss it might be something like this: The "science forums etiquette" thread in the Forum Announcements section offers the following advice: You may be intending to become a moderator by impressing the forum staff (with) your superb skills, impressive vocabulary, witty sense of humor and ability to make derogatory comments to newbies. That's all fine and good, but do it quietly and don't annoy the rest of the forum members. What is that all about? That's all fine and good? That's what the staff do? Don't annoy the members but the newbies are fair game - is this a fraternity? I suppose most on-line forums do tend to have these characteristics. There's bucket-fulls of research and discussion about how and why on-line communities tend to become echo-chambers for one set of opinions and cliques will form. Not much point worrying too much about it but it would help to have some idea of the historical development of this forum and hence some idea of what the current state of play is likely to be. Let's get some balance here, there is NOTHING WRONG with establishing a website for like-minded people to discuss things and share ideas. People can find another website to discuss their beliefs if they wish, or create a whole new one of their own. I found this website with a Google search and I was looking for a science forum, if I had wanted a place to discuss the inner beauty of the game of water polo I would have gone elsewehere. More-over I'd be grateful if the water-polo players kept their discussion off the Astronomy and Cosmology section of this Forum. There has to be some way to keep the threads and discussions coherent and on-topic. Can I just start by trying to understand the historical development of this forum and identifying what the current state of play is likely to be. Thanks for your time and best wishes to everyone.

No-one gave the website for live coverage. https://www.nasa.gov/content/live-landing-of-the-mars-2020-perseverance-rover

Hi Beecee and thanks for your article, I enjoyed reading that. Some evidence that dark energy is not uniform, thank you very much +1. Yes, total agreement on this. Even if there was some expansion of space, strong short range forces will dominate so that atoms and molecules don't tend to fly apart. The same applies upto scales of the local group where we can say the constituents are gravitationally bound together. It doesn't matter if space is expanding there, this can be overcome as you stated. I think this was discussed earlier on in this thread (Bufofrog, Joigus and Janus should get a mention for that). Sadly (for me, at any rate), it doesn't show that space IS expanding there. Worse than this it makes it hard to test. Bizarrely, it implies that any sensible co-ordinate system you set up in this region wouldn't give us any reason to believe that space was expanding (until you look outside the local group). Maybe the whole problem is caused just because we try and use co-ordinates that emerge from the usual cosmological models with the FLRW metric - but let's put that to one side for the moment (it's far too late for me to discuss anything like that coherently). You also mentioned isotropy and homogeneity of space. Yes, that's important and assumed on cosmological scales. However, space is not at all isotropic or homogenous at small scales. We can have some non-uniformity in dark energy, or some regions of space that aren't expanding the same way. Tired now but enjoyed the discussion, thanks to everyone, bye for now.

Hi Swansont, Thanks and best wishes to you. I think the main thing I wanted to get across was that the expansion of space may not apply uniformly. It applies on the largest scales of distances between distant galaxaies but not necessarily on the scale of lengths between atoms in a molecule. However, yes, since we don't have an adequate understanding of dark energy the typical form of the Einstein Field Equations with a cosmological constant may not apply on these small scales, if you want to look at it that way. Dark Energy can mean several things. Here's a few ideas (although I'm not suggesting you didn't already know them, just trying to open the discussion to anyone to follow): 1. In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales.... [Wikipedia - see below] 2. When we talk about dark energy, it might turn out to be a cosmological constant. Certainly, when we take all of the observations we have so far, it appears that dark energy is consistent with being a cosmological constant.... [https://www.forbes.com/sites/startswithabang/2020/12/25/ask-ethan-is-einsteins-cosmological-constant-the-same-as-dark-energy/] Yes, this would be the best way to look at it. The form of the gravitational field equations is determined by a set of basic mathematical and physical requirements, and the most general form of equation that fulfils these requirements just happens to be the Einstein equations with cosmological constant. It’s essentially just a background curvature that is there even in the absence of all other sources; the presence of such a background curvature modifies all other solutions obtained from the equation. 3. Two proposed forms of dark energy are the cosmological constant,[11][12] representing a constant energy density filling space homogeneously, and scalar fields such as quintessence or moduli, dynamic quantities having energy densities that can vary in time and space. [https://en.wikipedia.org/wiki/Dark_energy] 4. In an earlier post, I suggested more generally that we don't know much about dark energy at all and it could be described as "badly behaved gravity". I must admit I'm struggling to find an exact reference to where Neil DeGrasse Tyson called it "badly behaved gravity", maybe that's a term I only half-remembered. I can find references to Tyson calling it "Dark Gravity", "Unexplained Gravity", suggesting we don't call it anything, or that we call it "Fred" if we must. The essence of the idea remains - it is potentially misleading to give it a name which suggests it's a thing like a particle; or to suggest it's just a constant required in an equation. It is gravity behaving badly, something yet to be fully explained. ------ Now to connect with the later statement concerning bond lengths: If dark energy is adequately explained as a cosmological constant in the Einstein Field Equations then it is (fairly obviously) a constant that appears in the equations and it should apply wherever and whenever we apply those field equations to some region of space. In particular space would generally be seen to be expanding and this applies fairly uniformly from the scale of the lengths between distant galaxies right down to the lengths between atoms in a molecule. (There is an exception*). However, if dark energy is a particle or some field with values that vary in space and time, then the effects of this dark energy are not uniform. In particular, it is quite possible that the spaces between atoms in a molecule don't show the same kind of expansion that the universe as whole exhibits. We have astronomical observations that distant galaxies are receeding which leads to our requirment for the expansion of space but these observations can be satisfied by a space where most of that expansion is happening between the galaxies rather than within them. *We can even accept the cosmological constant and still attempt to exhibit small-scale situations where the expansion of space doesn't appear. It's already been mentioned that the field equations are NOT linear and so the cosmological constant doesn't always force a solution to the field equations which appears to have anything like a time-dependant scale factor multiplying the spatial components of the metric.

Ummm.... I'm not sure what you're asking. I try to have an opening that just says "hi" or otherwise sets a more reasonable scene for discussion. You could take it out if you wanted but the rest of the post may seem a bit too hard and direct then (and I really wouldn't want you to do that). Statement 1 was intended to soften the impact of the rest of the post rather than making it appear to be an outright disagreement. I do like General Relativity and would like to suggest that I am not the enemy or deliberately antagonistic to any view that GR should be applied to all situations. Statement 2 was put there to indicate that, whatever we may like to do, that doesn't mean that we can or should do that thing. In particular, I'm not sure that space expands everywhere in the same way as that indicated by the Hubble law. In particular, it may not apply over these small scales (bond lengths).

Hi @Markus Hanke, hope you are well. I like GR as much as the next person and I'd love nothing more than Einstein's Field Equations to apply everywhere with a cosmological constant - but is that enough to overlook the level of ignorance we have about Dark Energy? What makes you so sure that the Einstein Field Equations with cosmological constat applies to such small scales as a bond length within a molecule? We don't have an exact value for [math] \Lambda [/math] but just an approximation based on Astronomical observations and this includes adjusting the proportion of dark matter to obtain a good fit to the observations. To the best of my knowledge it isn't a full picture or a perfect fit to the data but just the best we can get. Worse than this, Dark matter is much more likely to be some sort of particle we may identify and then if we obtain some measurements of the amount of it in the universe, the value of the cosmological constant would need some adjustment again. If our estimated proportions of dark matter and dark energy tell us anything at all, then it's got to be that most of what's out there isn't what we thought it was. Don't get me wrong, I'd hate it if it was true - but the statement made earlier "perhaps it's a sign that our model of gravity is breaking down" is something we have to consider.

@iNow I don't want you to feel under pressure but if this isn't good, I'm going to cry all night. Then I'm going to come back here and change my +1 to -infinity. Seriously, thanks for the suggestion. I'm going to give it a try.

Skipped through the rest of those (two) videos. It's just a short survery or lunchtime interest thing over two lectures. Dissapointing, no graphics, no Mathematics, very brief. Might have been useful if you were there and could have asked questions but as a YT video, less useful than watching a good documentary. Still seeking recommendations for a good Cosmology lecture series.

Hi @Airbrush Hope you are well. Yes, although the emphasis probably needs to be put on "dark energy" and we might just as well replace it with Neil deGrass Tyson's favourite description of it as "badly behaved gravity". We don't have a good understanding of what it is or where it is. Perhaps it's not even a physical thing but just a requirement that the metric is found from an equation with a suitable cosmological constant. Finally, perhaps it's a sign that our model of gravity is breaking down. It's badly behaved gravity that is yet to be explained. We are reasonably certain that, on the largest scales, space is expanding. We are considerably less certain about whether space is expanding everywhere in the same way. On the smallest scales, is the space between two atoms in a molecule expanding? I can find references on both sides. Whatever the case, we can just point out that the dark energy problem isn't important on small scales, or if you prefer, that gravity isn't always that badly behaved - our local group is gravitationally bound exactly as we would expect. If dark energy was some kind of particle, then maybe it's just a case of checking to see if there's a particle of dark energy in-between atoms of a molecule, or thousands of particles in-between galaxies (but that would be too easy, so we can be fairly sure it won't be anything like that). Also, an earlier version of your post asked a bit more about what happens when galaxies collide - Joigus put a nice simulation up on this thread about 1 page back.

WANTED: Recommendation for a Cosmology Lecture series. Hi everyone. Not sure where to locate this post. Can anyone recommend a more up-to-date lecture series I can watch with similar detail and presumed background knowledge? I've seen this one and I can recommend it to others. Full lecture series on Cosmology from standford University available on Youtube (from 2013). https://www.youtube.com/watch?v=P-medYaqVak&t=1549s There's another full lecture course in the MIT opencourseware series on YT (from about 2014): https://www.youtube.com/watch?v=ANCN7vr9FVk&list=PL4BUyFYOFtUZIUtcSDo6vm4HPxS9aZmHC I've started watching that one. However, I can't help noticing that half the things mentioned as hopes for future developments have now actually happened and we have those results. Not sure if it's worth the hours of watching if I've already seen the Stanford lecture series and I'd appreciate any advice just about that. Can anyone recommend a more up-to-date lecture series please? Not a popular science documentary, something similar to a full lecture course for undergraduates or post-grads, please. Thank you for your time and advice. Late editing: Some statements made in the MIT lecture are becoming very painful to hear now (e.g. Energy conservation as a sovereign principle). Perhaps this isn't even as up-to-date as 2014, maybe it was just released to youtube on that date. I don't think I will be watching any more of this.

Hi again. I've been watching this thread for a few hours now and I can't help but feel that some earlier responses (including mine) have come over fairly negatively. For example, I'm really sorry that this thread was moved from Astronomy and Cosmology to Speculations. That is usually a decision made by ? whoever runs the forum but usually there is some explanation given for the move. Speculations does not mean "rubbish", there is actually a "trash can" section in this forum if someone felt it had to go in there. Presumably the post just seemed to meet the criteria for speculations and that's fine, do not let that worry you (it is by definition your speculations about a model for the universe). Additionally, getting a good roasting over your ideas is actually quite common practice in Science. Don't worry about that either, take it as a compliment that someone bothered to read the thing. Rest assured most of the people here have had similar experiences - we usually say "scrutinised by peers" but it means getting a good roasting from others. Finally, in direct response to your question - can you help me? Some people have taken the time to read what you've written and given it some evaluation, if that counts for anything. I've also got some more housework to get on with shortly which means that you've actually been the one helping me with a pleasant distraction, if that counts for anything. Genuinely, best wishes to you and bye for now. P.S. If you wanted a slightly different "smaller bit" to focus on: Looking at your diagram it seems that you were implying dark matter is like ordinary matter in some parallel time evolution that has some interaction (an inward push) on our universe. How would you explain that interaction or how would such a thing be detectable? Just focusing on a small idea like this would be a huge development on it's own.

Hi IRW03..70, hope you are well. I can do some humouring. Firstly, it's obvious you have spent some time thinking about this, getting some diagrams, some completely original ideas and some existing ideas from science all brought together. That's good. There's always a place for new ideas and we should certainly be encouraged to let imagination run wherever it wants sometimes. Let's not be too general but instead look at the specifics of your work - there are a few things you have mentioned that are really, really very good if not revolutionary. Congratulations. Additionally, a forum, like this one, seems a sensible a place to discuss your ideas. Now, it's not a case of presenting the bad points as such, I'm not suggesting that there are ANY bad points - this is more about presenting some balance. There is actually no shortage of ideas. More-over, by random chance if nothing else, some of these ideas will be extremely insigthful and extremely useful to further our understanding of science. Which means that sometimes, a really extraordinary idea has already been dismissed or failed to attract the attention it should have done AND this is terribly sad and a great loss to everyone. The problem is the amount of time it takes to go through everyones ideas and whether that would actually help, our society, to identify the useful and insightful ideas frequently enough. One Physicist could easily miss something or fail to identify the value if it if it's not in their area. How many scientists and how many hours should be used to survery each idea? Is this an efficient use of scientists, compared to what they are already doing? Missing some brilliant idea is a very high price to pay, I agree with that, but it may be the only practical solution to bring about the greatest production of science over-all. Would you consider reducing the scope of your work and presentation slightly? For example, the idea that there may have been two big bangs seems sufficient to start with and you'd get double the number of replies if you made a forum post just to discuss that. There is time to build and develop a bigger theory later. What have you got here: You have stated that you do not have the Mathematics to communicate or model your ideas. This means that what you have are some words, some diagrams and some ideas. To many modern Physicists, this means that you don't really have very much here at all, I'm afraid. Many of the diagrams you said were not yours but have been taken from elsewehere, that's fine - we do need to connect with existing knowledge - but I'll just look at the original stuff here for a moment. This seems to be what you are presenting as original: Comments: 1. There was more than one big bang. Yes, that's quite possible. See for example these references: I) Conformal Cyclic Cosmology - Roger Penrose. [Sorry, this is not a specific reference to just one article or paper but instead loads of stuff by Penrose related to this and you'll have no problem finding it yourself] II) Steinhardt, P. J.; Turok, N. (2002-04-25). "A Cyclic Model of the Universe". Science. 296 (5572): 1436–1439. arXiv:hep-th/0111030 - Paul Steinhardt. These two references (above) concern cyclic models of cosmology, in which there can be many big bang / big bounce phenomena. III) Any reference you like about string theory and membranes. A collision of two branes could be considered a big bang event and as such there is no limit to the number of these that could ocurr. 2. ...Preceeding big bang created spherical time.... OK, maybe. Got to ask if it matters at all. Time without space doesn't seem all that useful, is it important that time exists before space? Also don't forget that words like "before" don't really mean very much if time didn't exist. There isn't an absolute agreement on what "time" is. Do you really need time in your model or just some thing, some parameter that serves a purpose similar to time? Existing theory, like General Relativity, doesn't necessarily imply that something like "time" started at the big bang. It's just that a co-ordinate time breaks down. It is quite possible that a thing, a parameter like time, existed but is it important - does it help to model or explain time as it exists now? Always be very careful about any model and any set of words that accidentally slip into the use of notions like "before" and "after" and "preceeding" when the thing we usually consider as time or co-ordinate time really does seem to have come into existence only at the (usual) big bang. Additionally, this seems to be running against current trends. That's fine, just thought I'd mention that it's not current. More recent theory considers the possibility that "time" isn't required and there was only space. So it is only space that needs to exist first. Time then appears as what is described as an emergent property from the geometrical structures you have. Example: Big structures (like a cup of coffee) can have properties that weren't apparent in the smaller constituent parts - a coffee can have a nice swirl of cream on the top but this characteristic wasn't apparent in the molecules. A swirl is not a molecule or a single type of interaction between molecules and you require a million molecules of different types before a swirl can exist - but once you do have a collection of such molecules a swirl is a perfectly well defined emerget property it (the cup of coffee) can have. ---I've probably bored you already. I've also got tedious stuff to do like the housework, so I'll end here. Best Wishes to you, bye for now. ---

Hi @beecee, Yes GR is definitely a significant contribution and I nearly said that. Not sure about Carl Sagan but that's just a geography thing. In my part of the world we had other science presenters. Hi @Janus, Also a good suggestion. Thanks for quoting me but I think it was Wildie9 who started the new discussion - so I'll wave his flag. @wildie9 You may also want to read some of the above.

Hi again. Thanks for explaining your requirements. Sadly, I'm not an expert on higher order logic and it seems unlikely I'll have time to become familiar with it over the next few months. So I'll be bowing out of this conversation. Sorry I couldn't help. Best wishes to you, bye for now.

Hi again. You could start another thread for what is another topic - but it's your call. Greatest contribution to cosmology? The Hubble Law. Obviously, just one opinion.

@slomobile I'm still not really sure how to use the messaging system. If you need anyhting else write something here. Otherwise, best of luck to you with your work, bye for now.

Hi. Will this do? But it can be annoying anyway. Maybe best to leave it as a static link in text. I have limited ability to explain how you make it work, I just right-click on the embedded object and fiddle with settings and stuff until the post looks ok.

Hi Markus, hope you are well. In this context, I would define an amount of space (between the earth and the moon) as being something that is measured by the distance between the earth and the moon. Agreed. I said... The FLWR metric doesn't have to apply to such small regions as the bit of space between the Earth and the Moon... and maybe that should have been "doesn't" instead of "doesn't have to". The rest of the paragraph did discuss the better metric to use for that region of space given the exact distribution of mass that it has. Agreed. Space is not exactly like a river flowing and it's not exactly like a hamster cage either.

How does science model it? Well, that one can be answered. The usual model of the universe (on the really big scale) is General Relativity with the FLWR metric (given various names but LeMaitre, Friedmann, Robertson and Walker contributed and at-least a couple of these names get used). All that's important is that the spatial contribution to any proper distance measurement is multipled by something called the scale factor and this grows with increasing time. How you explain or visualise that is, as you say, largely up to you. It is a mathematical model and not any other sort of model. In particular, it doesn't say where some extra space comes from but only that there is more space as time evolves. Depending on how you look upon it, it either side-steps the issue or else indicates to us that this doesn't even have to be an issue. There are other areas of science that do make some attempt at explaining this phenomena, if you were interested, but for general relativity it doesn't need explaining, it just happens. Does the amount of space between the Earth and the Moon increase? Well I suppose there's three reasonable answers (two I can think of and one more because I'll probably have something else after I finish writing the first two): 1. Maybe not. The FLWR metric doesn't have to apply to such small regions as the bit of space between the Earth and the Moon. In practice, that small region of space is better modelled with the Schwarzschild metric using the Earth as the dominant mass. If you want to get a better metric you can create a customised metric by solving the Einstein Field Equations directly given the distribution of mass you have (let's say having the earth and the moon as perfectly round spheres of uniform density etc.). These customised metrics usually all tend to be just approximations because the equations you have to solve to find the metric are just too complicated to get exact solutions. However, the main point is that the FLWR metric (the one we use for very large scale universe models) is itself an approximation that is reasonable ONLY on the large scales. There almost certainly are some regions of space that are not expanding as fast as the FLWR metric would predict but then there are some regions that may be expanding a bit faster. On the very large scales, the FLWR metric is a reasonable metric to use but it never claimed to be an exact solution for that tiny region of space. 2. Yes it does. Here's a quote from the NASA website: The Moon is slowly moving away from Earth, getting about an inch farther away each year. ( https://solarsystem.nasa.gov/moons/earths-moon/in-depth/ ) However I should make it clear the main reason to explain this does NOT involve the expansion of space. None-the-less it is quite possible the expansion of space causes a tiny portion of that. 3. Umm....... Nope, can't think of anything else at the moment.... I guess point no.3 was a spare. Best wishes to you. Bye for now. ----- Wish I could separate these two replies here ---- I like the waterfall image, thanks. I'll check the paper later, thanks again. Found this YouTube video with a good waterfall representation of a Black Hole: Brian Greene, daily equation. Jump to about time stamp 4:40 for the waterfall.

Hi Zapatos, hope nobody minds me joining in. Well, yes you could think of space as "flowing" past an object that was bound in place by gravitational attraction to something else. I've seen several animations of space that do exactly this - show a steady flow of space as if it is something like a fluid flowing. That's ok. Alternatively, you can think of the expansion of space in a different way. Instead of space being stretched or any part of it being made to "flow", you can imagine that it is a reasonably fixed thing that can't stretch at all - there is just simply more of it (more space) being injected in to the universe all the time. In this view, nothing has to flow past anything else.... it's just that more stuff, more space is appearing (as if from nowhere). We had a hamster once and my children wanted this amazing hamster "home" that was actually more like a hamster universe, it had tunnels that connected various little habitat areas, up ladders, hamster wheels. Let's get a picture of something like this: I used to feel sorry for the hamster because he lived in an expanding universe. Whenever the children had a birthday or Christmas, they wanted another expansion for the hamster universe. Some of the tubes you could buy were of the stretchy type.... you could stretch these... but this kind of tube really is the kind of thing that models the universe as something that stretches and I suppose you could imagine that there was something like a flow of the tube material as you stretched these tubes. The more interesting extensions occurred when a whole extra section of tube was added. Some of the tubes were not stretchy, fluid-like things at all but were quite rigid - they had to be because you could build them straight upwards and support a little habitat area on the top of them. The rigid tubes were in short little sections of about 5cm length. Sometimes when the hamster was asleep, we injected more space into his universe... we didn't stretch any tubes or make anything flow past anything else... we just uncoupled a section and added in another section. Sometimes we didn't even have to make the bigger habitat areas move at all, they didn't have to flow or move across the floor either. We could replace a straight section with several curvy sections making a zig-zag if we wanted to. Anyway, that's the thing in this hamster universe - space didn't always stretch or flow past anything else.... sometimes new bits of plastic tubing were just inserted in-between the existing pieces of tube. Must have seemed amazing to the hamster.... new bits of space just being inserted in-between the existing bits of space. But all of this is just a model, you can imagine space is some sort of fluid that flows, if you prefer. Space certainly doesn't seem to be as empty as we once thought it was. There seems to be quantum fluctuations and energy even in a vaccum - but that's a different story.

Hi again @confused2021. I think I wrote 0.02 here when I meant 0.002. Anyway, hopefully the spirit of the discussion is solid even if the typing isn't. Also, isn't it great to know that other people can make mistakes? Got to look on the bright side. Also, @Bufofrog , I wish I had seen your reply first, would have saved me a lot of time typing. (I'm also secretly hoping that you'll run out of time to correct your spelling of Miles <--> Moles, then we can all have a typo).

Hi. I've only had time to scan through what you've written. Thanks for putting your own answers in here, we can see what you have done and what efforts you have made, that's great. In question 1, what you've done is reasonable but does seem to have a slight error. This could be just an arithmetic or trivial error, in that case.....well, ok, we all have those. BUT it could be a conceptual misunderstanding which we can do something about. We can also suggest a method to check your conversions make sense and reduce the number of accidental errors. A Molar concentration is a concentration in Moles per Litre. (I think you did know that but if not, you need to know it). The question has made things interesting by NOT making everything happen in 1 litre. They said ... dissolved in 5 mL. (I think you did realise this). When making conversions, it's a good idea to think about whether your final answer should be BIGGER or smaller after the conversion. You have correctly worked out that there will be 0.002 Mols of the glucose in 5 mL. Now is 1 Litre more or less than 5 mL? It is more than 5 mL, right? Quite a lot more than 5 mL. So there should be more Moles of glucose in 1 litre then there was in 5 mL. You have multiplied by 0.005 in this step, which will actually make the number of moles in 1 litre smaller than the number in 5 mL. That can't be right, can it? Can you adjust that conversion and then follow through with the conversion of Moles to milliMoles exactly as you did afterwards... and follow through for the micromolar units.... and the rest of the question paper. Question 2 is ok (BUT please use the corrected value from question 1). There is another way you could do this, if you want a method to double-check your own answer. Stick with what you have actually done, by the way, I'm sure that's what the teacher was looking for and that's what will be on the mark scheme. It's sensible to make things as easy to follow for the person marking this as you can. This alternative method is just a double-check for yourself. It just so happens, that you only dissolved the glucose in 5 mL to begin with. When question 2 says... take 5 mL of this solution.... that is ALL OF IT. You already worked out the number of moles you had in those 5mL half-way through question 1 (that was 0.002 mols). So the question is equivalent to saying what concentration will you get when you have 0.02 Mols in a volume of 500 mL (and make sure to convert to microMolar units at the end). Question 3: Break this into small steps. Step 1: Think... What do you want? The number of Moles. What do you Know? Information about Volume (you took 10 mL) and........... that's not enough... what else do we know?.... it came from the solution we knew about in question 2. What was the main thing we found out in question 2? The concentration of that solution. Is that enough information? We have VOLUME, CONCENTRATION and we want NUMBER OF MOLES. Are they related somehow? HINT: This is using the same idea you used (sadly, incorrectly) in question 1. There is a formula connecting these three things, do you know what that is? Step 2: You have a simple formula (mathematical formula), use it to find the thing you want. Remember to check the units you have used and the units they want in the final answer (like NANO-molar concentration) - do any conversions required. Good Luck, ask for more help if required.

Hi again. Godel has been mentioned everywhere so I assume you are aware of Godel's second incompleteness theorem. I'll para-phrase it here: Where P is the Peano arithemetic system, we have the result that the consistency of that system cannot be proved (unless P is actually inconsistent). This provides a fairly simple statement, ¬Prov("0=1"), in the Naturals which (almost) seems to meet your criteria. If that's all you wanted. See the references given earlier by wtf, I've tried to use the same notation. Why do I say ALMOST meets your criteria? It does not establish that P is consistent, we simply believe that it is. If P is consistent, then it satisfies your requirement. So the answer to your question is YES - we have a statement in the Naturals that cannot be proved or disproved. Conversely, if P is inconsistent, then we can prove anything, so the answer to your question would be NO - every statement can be proved. Additionally I'm not entirely sure what you mean by DP. You refer to the Dedekind-Peano-Axiom System which seems to suggest you are extending the Naturals to the entire Real Numbers. If this is the case, then the Godel numeralisation technique used in his original proof of the incompleteness theorems may not apply. As I understand it, we are required to find a unique numeralisation for every object (including numbers, field operations etc.) under consideration. This is not difficult for the Naturals as constructed by Peano's axioms since we need only numeralise the object 0 and the successor. However, finding a suitable numeralisation for the reals is obviously much harder. In particular we seek a 1-to-1 correspondance from R (the set of reals) into a subset of N (the Naturals), which defies what we understand about the cardinality of these sets. You may be correct in your original post but I have to say that you have me at a disadvantage: I was NOT aware that Godel's incompleteness theorems provide an example of an undecideable proposition in this case (if you have a link to this result, I'd be grateful for that) Hmmmm...... I'm aware this is becoming confusing perhaps I could make my point this way - Did you mean to write the following: If that's the sort of question you intended to ask, then it's difficult. We know that sometimes a Natural number can arise as the result of operations on irrationals, for example [math] \sqrt{2} \times \sqrt{2} [/math] is a perfectly good natural number. So we cannot exclude formula involving irrationals and just restrict our attention to statements provable or unprovable in P and the Godel numeralisation technique does not seem to generalise to the entire reals. Based on earlier replies you won't accept the easy way-out by exploiting the general weakness of most logic systems: For example, informally constructing a statement that just mentions Natural numbers in passing (like "this statement cannot be proved from the axioms AND 2+1 = 3") doesn't seem to be sufficient. Instead you seem to be putting some addditional requirments on the statement: Is it that the proposition "this statement cannot be proved from the axioms" requires formal construction within the logical language, or is it that you have very exacting requirements on the symbols that can be used? (For example only natural numbers and field operations like +, x ?). Apologies for length and lack of coherence, I am tired and off to sleep. Best wishes to you.

Puzzlement: This is extremely unusual. The language used by Tristan L is sometimes extra-ordinarily precise but it is frequently unusual. Some examples: "ownships" used in the OP instead of properties; A hyperlink to witcrafty that seems almost irrelevant. On brief inspection witcrafty seems to be a website about the formation of words. Brief inspection of some other posts Tristan has created or contributed to include one about entropy where an alternative way to patition microstates was presented that seemed to involve runes and the ability to communicate with runes. Other recent posts on this forum include one about notation studies (not by Tristan but by another new member, whose background is unknown). Speculation: @Tristan L are you more interested in notation than anything else? Are you trialing software that analyses and then utilises notation, perhaps software developed in some ML environment? What was the witcrafty hyperlink about? I can't understand how to turn off tracking cookies and disable other features of that website, so I haven't examined anything in it too deeply. None of this essential, it's probably just me overthinking the situation. Best wishes to you. Bye for now.