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About wtf

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  1. > I need to generalize my <* ordering. It needs to be an ordering on the whole set R because it has to be the possibility of any ordering. What does that mean, it has to be the "possibility of any ordering?" It's not even clear what you are trying to do. In order to make progress my suggestion to you would be to force yourself to write more clearly, one line at a time, and make sure each line makes sense. You don't have to write here, do it for yourself. Your ideas are jumbled because your prose is jumbled. That's why they make you write proofs in math class. Clear exposition leads to clear thinking. > wtf, do you mean that Cantor rational number thing? Yes, the fact that the natural numbers can be placed into bijection with the rationals. > I dont know why u asked this. Because you have to walk before you can run. You are trying to investigate or understand the relation between the usual dense order on the reals, with the well-order on the reals. Now the reals are a very complicated set. There is a much simpler set we have lying around that has the same phenomenon. You should study it to try to gain insight into the analogous problem in the reals Specifically, the usual order on the rationals is dense, just like the usual order on the reals is dense. Now you want to find a well-order. We know we can re-order the rationals so that they are well-ordered. This is a perfect analogy for what you are trying to do. That's why I suggested that you study it. It's so you can gain insight into the analogous problem for the reals: How to relate a well-order to a dense order on the same set. It's a standard technique in math, when you are working on a hard problem, to look at simpler examples. > I was going to show what I think is a way to reorder any set of real numbers in a dense or çontinuous way But why? Isn't the usual order already dense or continuous? You don't have to work very hard to find a dense order on the reals, the usual order is dense. > Im going to take a break from this now.This kinda gets to me after a while. I hope you got your money's worth from the convo. You should definitely take another run at learning about the ordinals. The ordinals are all about well-ordered sets. If you are interested in well-orders you are by definition interested in the ordinal numbers, even if you don't know it. I'll check this site from time to time in case you have more questions or comments. All the best, nice chatting with you. By the way I have a really cool book recommendation for you. Infinity and the Mind by Rudy Rucker.
  2. Here's another example you should be thinking of. We know that there's a bijection between the naturals N and the rationals Q. The usual order on Q is dense and the usual order on N is a well-order. You should try to run your ideas through this example since it's a countable and well-understood parallel to the case of a well-order on the reals versus the usual dense order.
  3. Just because t < a, why should we still have t <* a?
  4. Just because t < a, why should we still have t <* a?
  5. Run your argument on the naturals ordered by the usual <. Is there such a subset S?
  6. I say again: Please state in one sentence what you are trying to claim or show. * You agree what AC implies the well-ordering theorem. * You seem to understand that even in the absence of AC, the reals might still be well-ordered. You keep diving into your argument without saying in one clear sentence what it is you are trying to show.
  7. > Back in the 1980s I showed what I thought was a WO of [0,1] to a math prof at the Univ of Denver. He walked over to his blackboard, picked up a piece of chalk, thought for a few seconds and wrote out two infinite strings of numbers and asked me which one was greater with my order. I said uh oh and left. At least he didn't call your idea a trainwreck! Much nicer individual than me > A possible flaw in my hypothetical ordering is that I might be thinking of it in a usual sense. But, there just seems to be no other way. Any way you look at it you run into infinity somewhere. Again, I say all I need to do is show there is one subset that no l.e. can be found with the assumed WO to prove my assertion. I don't understand your assertion. Can you write it down clearly in one sentence? Are you saying no well-order of the reals exists even in in the presence of AC? That's false. Are you saying no well-order of the reals exists in the absence of AC? That might be false too. You seem to understand both these points. So I don't understand what you are trying to prove or what you are claiming.
  8. > I hope he will post something for me. That's a very kind thing to say. I agree that I deliver criticisms that could be taken personally, but they are not intended like that. When I say your argument is a trainwreck, that's a statement of objective fact based on considerable experience with proofs in this particular area of math. If I'm your TA I'd write "trainwreck" in big red letters. My intention is to be of assistance to you in understanding these ideas and learning to express your mathematical ideas more effectively, if that's your desire. To do that a certain level of precision is needed, both in concept and exposition. And I'll call you on lack of same. > Then wtf tells me I proved my assertion for dense sets and nothing else. If my proof is valid for these why does it not prove my assertiion? This makes no sense. Oh how funny. Yes I can see that you misunderstood this. I have a perfect explanation for you. I told you to carefully write out your proof that a dense linear order can't be a well-order, for the purpose of you getting practice in expressing your mathematical ideas and writing formal proofs. I reiterate that suggestion. I did not mean to imply that this would help your cause. We agree that it's not sufficient, because you still have all those OTHER linear orders that aren't dense but that might be well-orders. But you would learn a lot from struggling -- and yes it is a neverending struggle for ALL of us -- to force yourself to write math clearly. That is the only way to get better at it. I suggested this as an exercise. I hope that's not presumptuous in a message board format. I've been through this material and I've had proof-writing beaten into me by distinguished professors at some of the nation's finest universities. It's simply the only way to learn this stuff. If there were a pill we could take, we'd all take it. As Euclid said: There is no royal road to geometry. Even the King has to "do the math" if he wants to learn. > I guess the natural numbers being well ordered is supposed to contradicts my proof. You are asking why I talk about N. The reason is that they are the SIMPLEST example of an infinite well-ordered set. So when we're trying to get insight into a well-ordering of the reals, one place to look for intuition is in the much more familiar well-ordering of the naturals. And this is a math trick! Whenever you have convinced yourself that you proved something about a complicated object, run the proof on a simpler object that the proof still applies to. If you end up proving the naturals aren't well-ordered, then you know your proof's wrong. Re ordinals: Your interest in well-orders has led you back to the ordinal numbers. They are hard for everyone at first but they are one of the very coolest things I know about in math. They're well worth another run at that part of the book. By the way a well-order of the reals is an uncountable ordinal. That's a very strange beast to get one's mind around, it takes some work and study. So if you want to think about well-ordering the reals, you need to learn about ordinals and work your way up to the wild idea of an uncountable ordinal. The existence of an uncountable ordinal can be proved WITHOUT the axiom of choice. That is a very interesting fact I think. I can walk through the proof but you do need to believe in the ordinals first. > Iĺl get pushback from wtf for this. Didn't you initially ask for someone who knows this material? Be careful what you wish for > wtf mentioned other models of set theory that allow well ordering without the need of the AC. I truly want to see these. Yes this is not a hard reference. It's a mention of a fact that, once I thought about it, must be true. I found it in a very unlikely place, in the Talk page for the Wiki article on the Banach-Tarski paradox. Whenever I look at a Wiki page I often look at its Talk page, where the editors argue about what the article should say. You get a lot of insight from these sometimes. So, here is the Talk page for the Banach-Tarski paradox:–Tarski_paradox And way down the page there is this conversation about how much Choice is needed. (Sometimes you don't need the full axiom of choice, you can get by with weaker principles like countable choice, dependent choice, and some other intermediate axioms.) Without going into this in detail -- readers can click if desired -- someone at some point said this: And it must have been known since the sixties that it's consistent with ZF that the reals can be wellordered but some larger set (say, the powerset of the reals) cannot. The moment I thought about this I saw it must be true. The well-ordering theorem says that ANY set can be well-ordered. Its negation says that SOME set can't be well-ordered. But it doesn't say anything about which set that is. So it must be logically consistent that the real numbers are well-ordered, yet some larger set isn't. In this particular model of set theory, well-ordering is false and the axiom of choice is false yet the reals are well-ordered. These are deep waters.
  9. Are you talking about the proof that the reals are uncountable? Or that the unit interval can be placed into bijection with the unit square? It's the proof of the latter that you outlined.
  10. Yes Cantor did this. See the section titled "Squared Infinities" here ...
  11. > OK, I see what I did. That a set can be WO depends on the AC. The two are equivelent statements. The natural numbers are well-ordered in the absence of AC. And as I noted, even the real numbers might be well-ordered even in the absence of AC. What AC says is that EVERY set may be well ordered. > What I showed is WO does not exist naturally on its own. Its like saying if we can live on air we do not have to grow food and we know the first part is impossible. That doesn't make much sense. Even in the absence of AC, there's a model of set theory in which the reals are well-ordered. And without AC, plenty of sets are "naturally" well-ordered, like the natural numbers, or any finite set, or any of the transfinite ordinals. You certainly did NOT show that "WO does not exist naturally on its own." You didn't show anything like that. > What I did was just an exercise. You don't seem to have understood the flaws in your own thinking. You think you proved that "a WO does not naturally exist on its own." First that doesn't make sense, since you haven't defined your terms. And second, under any reasonable interpretation, it's false. > Of course something like this could never overturn 100 years of history. Of course it cannot be that simple. If a set is WO then there you have your set of numbers for the AC. No no no no no no no. Even in the absence of AC, all the ordinal numbers are well-ordered. And even in the absence of AC, the real numbers might be well-ordered. > Yes, its clear the natural numbers can be WO, but my thesis was that not every set can be WO. If you accept the AC then of course you say they can. "I say" they can? What do you say? You deny that AC implies that every set can be well-ordered? Your phrasings are so imprecise that they are causing your thinking to become confused.
  12. @taeto has already addressed and falsified your latest idea. I'll just add that you can get some clarity by running your argument on the natural numbers {0, 1, 2, 3, ...}. The naturals are well-ordered and 0 is the first element. So let T = {1, 2, 3, 4, ...}; that is, the naturals with 0 removed. Now what is the least element of T? It's 1. That's how well-ordered sets work. Wouldn't your argument "show" that the naturals aren't well-ordered? As @taeto already asked, why do you think T doesn't have a least element? > Oh, and the Continuum Hypothesis is also in these equivallent statements. No no no. CH is independent of all of them. In fact if you assume CH you get a nice well-ordering of the reals that you can almost visualize. The first uncountable ordinal can be proved to exist even without AC. If CH is true, the first uncountable ordinal well-orders the reals. And with a bit of study, you can visualize the first uncountable ordinal. It's just the set of all possible well-orders of the naturals. Equivalently, the first uncountable ordinal is the set of all countable ordinals.
  13. I ran across something that I didn't realize before, but it makes perfect sense. The well-ordering theorem is equivalent to AC. The well-ordering theorem says that ANY set can be well-ordered. That implies of course that the reals can be well-ordered. However, suppose we deny AC. Then there is SOME set that can't be well-ordered ... but it might not be the reals. That is: There is a model of set theory in which AC is false, and there is SOME set that can't be well-ordered, yet the real numbers are well-ordered! So I was wrong before when I said that you can deny AC and thereby prevent a well-order of the reals. On the contrary, the reals can be well-ordered (in some model of set theory) even in the absence of AC.
  14. If you leave the thread as it is, someone curious about this subject might come by in a few years and learn something. I think your instinct was right. You did show that a dense order can't be well-ordered. The point wouldn't be to give up on your studies. The point would be to learn more. As I said, your instincts are good. I'm trying to encourage you to learn a little more, not to give up. Remember that when Wiles announced his celebrated proof of FLT, someone found a significant error in his work. It took Wiles over a year to patch the problem and get a completed proof. You did a good job. You had an insight, you proved something interesting, and you realized that you had a logic error. Wiles did the exact same thing! If you are rusty at math, well you are less rusty now than you were before. Keep at it and soon the rust will turn to shiny stainless steel.
  15. > I gave an answer using a library computer and didn keep track of the time limit. It was completely lost. Iĺl have to do it again. Sorry to hear that. On the other hand it's been my longtime experience that whenever I write a lengthy screed and manage to lose it, my second effort always comes out much better. > Iḿ thinking Iḿ sticking to my original statements. But your original statement is wrong. You showed that no dense linear order can be a well order. But you didn't account for the case of some non-dense linear order. > As I said before if a order relation, <*, WOs the set of reals there has to be a l.e. according to <* in every possible subset of the reals. Every nonempty subset. Picky picky! > If every possible subset cannot be constructed there is no guarantee a subset made up of elements each from each subset of the reals exists either. The powerset axiom guarantees the existence of every subset of the reals. These subsets do not need to be "constructed," whatever you might mean by that. I do not have to explicitly exhibit each of the subset of the reals to know they exist. But you seem to be arguing against AC. That's pointless. You're perfectly free to adopt its negation, that gives a consistent set theory. There's no argument to be had. If you reject AC then well-ordering is false and we're done. You lose a lot of modern math and a good chunk of physics but you're not wrong. Only out of step with contemporary mathematical practice, which I perfectly well agree is historically contingent. > I like this Chromebook yes and no. There are a number of limitations, but its easy to carry anywhere. There is no delete' key.You have to use backspace. I think I finally found out how to disable this auto correct. I'll check it out. Google already owns my life anyway. Resistance is futile.