discountbrains

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

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1 hour ago, discountbrains said:

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. And, it certainly could be that a<*r<*c<* b<*....In fact, any number of elements not in T could be mixed up in there, but that doesnt mean there is a z in T with z≤*x for all x in T. I might write a typical ordering of mine as   

....<* a<*r<*c<* b<*p<*... where r and p are not in R.

wtf, do you mean that Cantor rational number thing? 1    2      3      4 ...

                                                                                                           1/2 2/2 3/2 ....

                                                                                                          1/3 2/3 3/3....

                                                                                                          .       .         .

This, of course, is where you start in the upper left corner and count back and forth diagonally showing thereś a one to one onto relationship to the natural numbers and doesnt apply to what im saying. Clearly this is a WO set. I dont know why u asked this. 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 will do that later. Im going to take a break from this now.This kinda gets to me after a while. If anyone has any comments or questions I will still read them.

> 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. https://www.amazon.com/Infinity-Mind-Philosophy-Infinite-Princeton/dp/0691121273

Edited by wtf

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