# discountbrains

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1. ## Where to submit my proof that the set of real numbers can't be well ordered

Here's your problem: You have a problem with your counterexample. In order to demonstrate there is a well order relation for R you must have that my set, T, contains an element to be a min for T. Hence for any order you present to be a WO T cannot be empty! Its as simple as that. I actually should have said this a while back. Of course, there is an unimaginable number of examples of order relations which would give no elements in T. You have to present one that produces an element in T to be a min of T and also elements of all of its subsets likewise. Naturally anyone is on risky ground when they go against conventional wisdom like me. I will revert back to my original title of my post. I think I mentioned contacting the AMS journal and was told my work did not fit their requirements by their secretary-true its no multi page article. Wonder if she might even be another one of these low level gate keepers like are encountered many places. I have another issue with your example; won't bother mentioning it here. I'm comfortable with my argument now. You brought attention to potential flaws. There are other math societies. I haven't had any direct access to any academia for a long time. I will get someone's attention somewhere. My 'proof(?)' is what it is.
2. ## Where to submit my proof that the set of real numbers can't be well ordered

Yes, I know that's exactly what I'm saying: the reals can't be well ordered. And, you say there is a proof without the AC that says they can. Yet, I'm getting the result I'm getting. I think I can come up with many, many examples of your logic. I might say all my numbers in my set S add up to a whole number like n+m=p (p a whole number). For example 1/3+2/3 =1, 2+1=3... and you say what about 1/2+1/3 etc? And then say my statement is false. This may not be an exact analogy though. Let me reiterate, I believe my last version of <* must include numbers from T. Let me see if this is what u are saying: I say all the car prices on a Chevy dealer's lot are ordered a certain way and u say my statement is invalid because the Ford dealers are not ordered that way.
3. ## Where to submit my proof that the set of real numbers can't be well ordered

Yes, Neither 0 or 1 are in T. 1<*everything else makes any other x and y such that (x,y)∈<* not be in T. So T is empty with your order relation. You are pointing out an imprecision in my argument. That's good. I should restrict any order relation to be if x and y are in T, (x,y)∈<*. Or this could be written in other ways...... No, there are lots of well orderings for a lot of sets. I'm trying to show there are some sets that can't be well ordered.
4. ## Where to submit my proof that the set of real numbers can't be well ordered

I admit I should take more time for serious thought on your example, but will need to see if your example is meant to be totally unworkable with my definition of T. Yes, again I can come up with definitions of <* that will make T seem empty right off the bat. So what. If I say T contains all elements of S except a and b while a and b are also in S then that's what T contains. Again defining <* to make T contain nothing is beside the point because u must have at least one min in T to make T well ordered. You're trying to make T empty so u don't have to answer the question. You could offer to help me define T and <* like suggest I start by saying "if T is a nonempty set".
5. ## Where to submit my proof that the set of real numbers can't be well ordered

I see in my latest definition of T I made a couple minor errors. Since T doesn't contain a and b T can't = S near the bottom of my proof. Since my initial conditions state all other elements in S are in T clearly T is not empty. However, are you really a math prof? You know and you stated your example is not a well ordering. I don't care about this at all. This is entirely not the point. I care only about what is proposed to be a WO and about 6 lines down I conclude such an ordering implies, in fact, various sets I present must be empty. You do realize if u can produce an ordering that well orders my set thus negating my argument u actually have done something pretty profound. You will have actually produced a WO for the reals. You seem to be stuck the idea of finding anything to show my T is empty. Yet I get -32 votes here-amazing.
6. ## Where to submit my proof that the set of real numbers can't be well ordered

That's not what I'm trying to say. Clearly, to me I don't intend to cherry pick any order relation. I believe I clearly stated this applies to any order you choose. I should revisit my proof, but I should stipulate a,b are in S, not in T, and T contains all other elements of S. This leaves no doubt T is nonempty. Interesting some of my replies are missing. Could it be I used a popular slang word that's not demeaning to anyone? No, no, no, NO! You don't redefine my T!!! What is the issue here is that <* is arbitrary and you can define it any way like. Three times u redefined my set, T....... OK, got it. I will need to take a look at your example of <*. Maybe I can find a way around it. Wait a minute, who said the "everything else" is in the usual order? I really think I need not even require x<*b in T. All I'm concerned about is the lower bound or the min .
7. ## Where to submit my proof that the set of real numbers can't be well ordered

I did no such thing!
8. ## Where to submit my proof that the set of real numbers can't be well ordered

Here's another way to look at it: We don't hesitate writing a set like {x: a<x<b} with natural ordering. Isn't it natural to write {x: a<*x<*b} for a different ordering, <*? ....Lets suppose there is some dude in an alternative universe for which <* is the natural order of things and < is an alternative order? So, any of these sets will have plenty of elements in them.
9. ## Where to submit my proof that the set of real numbers can't be well ordered

I know what you're trying to say. Since you are starting with its given every set can be well ordered it must follow my sets are empty. It that not correct? Somebody here doesn't understand what's going on. First of all, I defined S=(0,1) with usual order not your set. Do u not understand I am presenting a set, T, which I showed can't be well ordered. You keep saying my sets are empty. Let me put it this way: we can think of any uncountable subset of the reals as an infinite deck of cards. We can shuffle this deck any way u like and get a new order. If u pick any card and claim there's no other card below it I can always find one that is. Or, lets replace any two digits of some decimal number with two others and then replace the two others with another two. We have a reordering. Or we might change all the digits of every number. Whatever number u say is the first I still can find another number before it. This doesn't matter how u order things. What u are trying to say is my set T does not exist by saying its empty.
10. ## Where to submit my proof that the set of real numbers can't be well ordered

Here it is. If you find any little mistakes let me know. For reasons I gave above how can we imagine the only sets that satisfy my definition above of T have to be empty.

12. ## Where to submit my proof that the set of real numbers can't be well ordered

z0 z1 z2 OK, here's my proof for your perusal: z0 z1 z2 The proof is by contradiction. Let S =(a,b) where a,b∈ R, 0< a <b. Let T= {x: a<*x<*b} where <* is a total order relation other than, <. I define S this way so all z∈S are in T. S is an denumerable set so T is also. If <* well orders T then ∃z∈T such that z0 ≤*x ∀x∈T. It must be that a<*z0 and {x: a<*x <*z0 } is an empty set z0 ≤*x ∀x∈T T..... I'm going to have to write this out on paper and scan it as an image. This is just too hard. I was nearing the end and it erased my work. I am proving by induction we can generate a denumerable string of minimums of every subset of T arising from after the previous minimum was deleted. This is a denumerable set and actually contains all elements of T. If we choose a z in T it is the min of any subset of T of numbers greater or equal to it. Therefore its also a member of a string of minimums of T. So, T is = to this denumerable string of minimums. T thus is denumerable. Contradiction.
13. ## Where to submit my proof that the set of real numbers can't be well ordered

Using the induction theorem might be a little tricky for a lot of proofs. One has to be careful. Here's what I think should be sufficient to show your example doesn't fit my set definition: Lets pick any arbitrary z in your 'E, everything else'. So, there could be some x<*z in E and clearly there are x's in E such that z<*x and we generate the set T={x: z<*x}. You must have a min, m, in T such that m <* or = all x in T and it has to be that z<*m and a set {x: z<*x<*m} must be an empty set (you maintain). Keep in mind T is all x in S such that z<*x. S might be R or (a,b)-with usual order-a subset of R, a and b are real numbers....... I think I know what you are saying. That is, how do I know my set (S,<*) can be constructed for any order relation <* or that {x: z<*x<*m} has to contain numbers? I can still build such sets from your 'everything else'. This has always been my premise from the beginning that for any possible order relation sets of the above form can always be constructed. I need to look up and see if the union of any collection of any countable sets is still countable and maybe use this result to help prove my claim.
14. ## Where to submit my proof that the set of real numbers can't be well ordered

I keep telling u over and over that the set I'm presenting is a set that cannot be WO. You keep presenting a different set. This is not my set. Do u think I can use the induction theorem implies your "everything else" does not include all x in S? I think I can; I'm putting my argument together to post.
15. ## Where to submit my proof that the set of real numbers can't be well ordered

Yes, I quoted you, but deleted your quote and then gave my response which made my reply confusing. What you are saying then is that you already assume S is WO and hence u can get the result it is WO. I'll have to think about your claim that the set we are arguing about is empty. If u can give an example set where u say its empty then u can deny my claim. you are giving an example of a set you concocted; I'm presenting a different set. Yes, that's what I have read about ordinals before: that each number is really represents the set of its predecessors. What puzzles me is what is the purpose of all this. Anyway, I'll think about all this.
16. ## Where to submit my proof that the set of real numbers can't be well ordered

You talked about ω + ω. This is the problem I'm having with this ordinals: How do you determine when one sequence ends and the next begins?
17. ## Where to submit my proof that the set of real numbers can't be well ordered

OK, what I said is not acceptable. There other ways to go about this like u said. What I said later about the functions is simply how derivatives have been calculated for centuries-nothing new here. It is interesting though to think of a function expanding or contracting instead of a specific value y=f(x).
18. ## Where to submit my proof that the set of real numbers can't be well ordered

Edit: I might say (S,<*) is S with a and b removed and = {x: a<*x<*b}.
19. ## Where to submit my proof that the set of real numbers can't be well ordered

Yes, I am just getting back to this. What I have done is use 'S" for both sets and defining (S,<*) like I did I actually made a and b the min and max of (S,<*). I don't want to do this of course. There are other straight forward ways of defining it. I'm thinking of moving on. Because as I realized long ago if you try to reorder a line segment [0,1] on the x axis you end up with a plane [0,1]x[0,1] of an infinite number of points in XxY or if you draw a squiggly curve you pass through the same numbers repeatedly which is not an order relation. So, what value is this anyway? This gave me another thought which is by drawing a curve f(x) on the y axis of x and if d(xn,xm) is a metric you would get d(f(xn), f(xm)) being greater or less than d(xn,xm). This might lead to another way to finding the max or min of a curve. That is where d(f(xn), f(xm))=0 with d(xn,xm) not =0 for some m and n. I wonder if this has ever been looked at.
20. ## Where to submit my proof that the set of real numbers can't be well ordered

Whoops, I hoped I would correct my mistake before someone else did. L might say S=(0,1) with usual order and (S,<*)={x: a<*x<*b where a,b ∈ (0,1) so I don't restrict myself too much where a counter example can be made. Lets suppose you exhibit an ordering that produces a min for my set I believe I can show it doesn't work for all sets. This may lead to a new theorem that for any set a ordering exists that produces a min for that set, but it won't for all sets. This is leading to a lot to think about.
21. ## Where to submit my proof that the set of real numbers can't be well ordered

" Why do you think 0 is the greatest lower bound? It's clear that 1/2 is a lower bound for S and is greater than 0. So 1/2 is the greatest lower bound of the set S. It's a) A lower bound for S; and b) Greater than every other lower bound for S" But, I stated at the outset S = (0,1) with the usual order. (S, <*) is strictly made up of numbers from S. 1000, -5/2, etc are not in S.
22. ## Where to submit my proof that the set of real numbers can't be well ordered

Surely u know why. Maybe you're looking for more precision. Precision is required in math. I believe all I need to do is state <* is just like the usual order, <, only its a different ordering. OK, how are arbitrary orders defined? I need to know this. Is wtf's ordering above acceptable? That is, 1, 1/3, 1/5,...1/2, 1/4, ... etc I've always wondered how u get from numbers in the first sequence to the next sequence. Might this be in the study of ordinals? In a case like his sequences the order might be as f(n), f(n+1), ... Here's an interesting order: How do they determine all the barcodes, UPC codes, for products in stores like Walmart? Some items in ones home might have to be partially ordered.
23. ## Where to submit my proof that the set of real numbers can't be well ordered

wtf made an error-a fatal error. He supposes 1/2 is the min of S. To be such 1/2 has to be in S ⇒ 0 <*1/2 <*x for all other x in S. Since 0 is the greatest lower bound of S, 1/2 is no lower bound and hence not a min of S. To uncool I'll have to think about if the way I defined S is sufficient or do I need all this other stuff which is unconventional and needs some cleaning up.
24. ## Where to submit my proof that the set of real numbers can't be well ordered

You meant S={x: 0<*x<*1} right? 0 is the greatest lower bound and 0 is not in S. There is no <* lower bound in S. To answer wtf if 1/2 is the min of S then 1/2 <* x for all x in S not = to 1/2. I never really wanted to do this, but lets go back to the very primitive notion of arithmetic-back to ancient Egypt or even the cave man. He might think the fish he caught is longer than all the others. He has to cut some notches in a stick to measure and compare each fish. We can order things anyway we like, but we are using math when we assign numbers to, lets say, a and b. We say a<*b if there is some way to get from a to be. We don't care how. We say a<*b iff we make a symbol, +*, such that a+*σ=b. We can continue with this idea and say σ=q+*q. It follows that a+*q <*b. So if 1/2 is in S we can write there is a q such that 0<*0+*q<*1/2 and of course 0+*q is in S by the definition of S. Therefore 1/2 can't be the min of S. σ is a number -σ is not defined for this. Lets say σ >*0 and σ > 0.
25. ## Where to submit my proof that the set of real numbers can't be well ordered

This was my original concept. The idea is given any order relation a set, S, can be described containing a number, a, such that for any number you choose, M, in the set not equal a, a<*M. The opposite notion to this is given any subset of R you can describe an order relation that produces a minimum in this set. The problem is the order relation might only produce this for this set. I know what you are saying. That I must produce every possible ordering <* and prove it doesn't produce a minimum M for S. I believe all I need to do is state what seems obvious that a set {x: 0<*x<*1} exists for any <*. Set theory allows us to produce any set we can think of.
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