February 24th, 2018, 02:04 PM  #41  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,404 Thanks: 2477 Math Focus: Mainly analysis and algebra  Quote:
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It's not up to you to define the real numbers. We already have several equivalent definitions of them. You can, by all means, come up with a different one but it must be equivalent to the definitions that already exist. Otherwise you are talking about something other than the set of real numbers.  
February 26th, 2018, 07:23 AM  #42 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,464 Thanks: 106 
For 4place decimals the count should be .0000, .0001,...,.9999. That makes the count easier to visualize for all n. The count (a PROPERTY) of nplace decimals is finite as long as n is finite. As n $\displaystyle \rightarrow$ infinity, the count $\displaystyle \rightarrow$ (countable) infinity. I believe the term "countable infinity" comes from Cantor. I distinquish between limiting sum of an nplace decimal, which is not unique but rather a property of nplace decimals, and the typographical limit as n approaches infinity, which is unique. Two real numbers are the same if every digit is the same for all n, not if they approach the same $\displaystyle \epsilon, \delta$ limit. The real numbers are unique as I have defined them. Function: DEFINITION: Map from elements of one set to another. Continuity: PROPERTY of a function. Limit: PROPERTY of a function. $\displaystyle \epsilon, \delta$ for f(x) or $\displaystyle \epsilon$, M, n>M, for f(n) [QUOTE=zylo;588830]REAL NUMBERs are defined uniquely by "infinite" (unending) sequences of natural numbers. The sequence IS the real number. [QUOTE=zylo;589093] Specifically, the limiting case as n approaches infinity (for all n) of a finite sequence of digits. 0,1,2,3,...,9, as in .135396782......, nplaces What is your definition of a real number? 
February 26th, 2018, 07:43 AM  #43 
Senior Member Joined: Oct 2009 Posts: 519 Thanks: 165  
February 26th, 2018, 08:52 AM  #44  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,464 Thanks: 106  Quote:
If you feel comfortable dealing with analysis with a mental image of cuts, fine. You mentioned hyperreals in another thread. How do you think of hyperreals in terms of cuts? But I can see that an abstract mathematician might think of real numbers as defined things, names. EDIT What is the $\displaystyle \lim_{n \rightarrow \infty}$ of f(n) in terms of cuts? In my opinion. cuts are a way to associate infinite decimals (real numbers, analysis) with points on a line (geometry). Last edited by zylo; February 26th, 2018 at 09:03 AM.  
February 26th, 2018, 10:46 AM  #45  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,404 Thanks: 2477 Math Focus: Mainly analysis and algebra 
I've compared your post with the one before and there's almost nothing new. This adds nothing at all  unless for all n is back to including infinite decimals as it has in the past. But then your list becomes meaningless because you can't even construct the second element. It does, but that doesn't mean that you can use it to describe anything that you guess is a countable infinity. It has a precise meaning that does not apply what you are saying. Quote:
Again, your definition is wrong. Or perhaps you intend to talk about a different set to the one everyone thinks of when you talk about real numbers. But in that case, why choose deliberately confusing terminology. This also is false. It is the limit of a sequence of rational numbers that is the real number.  

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