June 19th, 2017, 08:05 PM  #171  
Senior Member Joined: Jun 2014 From: USA Posts: 310 Thanks: 21  Quote:
$$x + x + x + x + x + \dots \text{?}$$ Your version of the reals under this definition lacks the Archimedean property: https://en.wikipedia.org/wiki/Archimedean_property I suggest you read an introduction to real analysis. It helped me when I was struggling with some of the concepts that you are. In particular, the reals exhibit the Archimedean Property (see page 5 = page 13 of the .pdf file: http://ramanujan.math.trinity.edu/wt...L_ANALYSIS.PDF ) "The property of the real numbers described in the next theorem is called the Archimedean property. Intuitively, it states that it is possible to exceed any positive number, no matter how large, by adding an arbitrary positive number, no matter how small, to itself sufficiently many times." Quote:
See Carl Mummert's answer in this stackexchange thread: https://math.stackexchange.com/quest...gofthereals "If you work in $L$ (that is, you assume the axiom of constructibility) then a specific formula is known that defines a well ordering of the reals in that context." Basically, people have pushed these concepts far beyond what you are doing here. You aren't going to get anywhere because you refuse to learn the basics. We can't keep explaining to you that 1  0.999... = 0 as opposed to "the least [positive] irrational number." There is no 'least positive irrational number.' It doesn't exist. Further, a well ordering of the reals does not require a least positive irrational number under the standard $\leq$ ordering. It only requires a least element under some different ordering. For example, 0.5 could be the first element in a well ordering of the reals followed by $\pi$. You still haven't grasped the concept of a well ordering.  
June 19th, 2017, 10:55 PM  #172 
Senior Member Joined: Mar 2017 From: . Posts: 265 Thanks: 4 Math Focus: Number theory 
Well then. The number 0.00..01 doesn't exist. Even if it exists it is irrelevant in this case. Also operations on such the number doesn't give meaningful results and renders it meaningless. Well ordering is simply about having a pattern from which all elements of the set to be ordered would follow. A standard ordering is an example of such a pattern. That's how I understand the concept and it might be I haven't yet really grasped it. A well order of all positive real numbers exists. It is impossible to write it down. 
June 19th, 2017, 11:33 PM  #173  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra  Quote:
Quote:
You need to understand that a wellordering of the set of positive real numbers does not use the use the normal ordering relation $\lt$. It uses a different relation under which we say that the first number we pick is the "smallest", the next is the next "smallest", etc.. Then, the only problem is to show that, despite not being listable (countable) we can (in theory) select the real numbers one at a time until they are exhausted. This is completely counterintuitive, but is facilitated by the axiom of choice. It's kind of barmy but logical at the same time  in fact that is what will send you to the mad house. Last edited by v8archie; June 19th, 2017 at 11:37 PM.  

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