July 6th, 2018, 06:47 AM  #41 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2643 Math Focus: Mainly analysis and algebra  
July 6th, 2018, 07:09 AM  #42  
Global Moderator Joined: Dec 2006 Posts: 20,747 Thanks: 2133  Quote:
If you don't like that terminology, just read on a bit to find that Maschke was referring to oridnals or cardinals that aren't finite. It's unreasonable to object to "transfinite" as doubletalk and then use the term. Induction isn't a limiting process, yet you readily state "in the limit" in a context where all you have is that something holds for each positive integer n. The wording "for all n" isn't "a little flaky", it's just an alternative wording. If you're unaware of a formal definition of "transfinite" or "transfinite", that doesn't imply that such a definition isn't possible. Quote:
What do you mean by "more points than you (or anyone) can count with a number system"? How would counting be done without a number system?  
July 6th, 2018, 08:57 AM  #43  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
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Transfinite is an imaginary concept which serves no purpose other than to throw a wrench into the gears while pretending to be logical, mathematical, and unassailable. None of the above. But it impresses the hell out of freshmen. I intentionally didn't say freshwomen they're not that easily impressed. Quote:
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Decimal or any radix definition works fine for anything I can think of in analysis or topology (completeness, neighborhood, nearness of points, etc).  
July 6th, 2018, 09:32 AM  #44 
Senior Member Joined: Oct 2009 Posts: 798 Thanks: 298  
July 6th, 2018, 09:35 AM  #45  
Senior Member Joined: Oct 2009 Posts: 798 Thanks: 298  Quote:
I'm fine with that, those are all respectable opinions in my book. But please don't refer to the number line as being the real numbers then.  
July 6th, 2018, 10:46 AM  #46 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
Seems we haven't defined a line. A line is all nplace decimals, or radixes, for all n. It can be pictured physically as the points on a light ray between two points divided into 1/10^n segments, for all n. It follows that m(1/m) = 1 for all m, m=10^n, and $\displaystyle \infty 0$=1. Edit. If you like, as the number of subdivisions becomes larger, the numbers (points on the line) become “real.” That’s about the best you can do without yoga. Last edited by zylo; July 6th, 2018 at 11:36 AM. 
July 6th, 2018, 11:59 AM  #47 
Senior Member Joined: Aug 2012 Posts: 2,323 Thanks: 715  
July 6th, 2018, 12:43 PM  #48  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
I know, I know, no matter what n is it’s a rational number. But that’s just semantics. As time goes by, your division construction fills up all the holes. And where is the line you use as a yardstick for your numbers? It doesn’f exist except as a line of molecules, so you have to define it.  
July 6th, 2018, 12:57 PM  #49  
Senior Member Joined: Aug 2012 Posts: 2,323 Thanks: 715  Quote:
If you mean something else by "for all n" please say what you mean. Quote:
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The "real line" is defined as the set of real numbers. As Micrm@ss noted, there are other philosophical conceptions of line. Are you referring to one of those alternate ideas? Last edited by Maschke; July 6th, 2018 at 01:10 PM.  
July 6th, 2018, 01:59 PM  #50  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
If it makes you happy, every time you divide the line 10^n times, divide it 10^(n+1) times. And of course the question, what is your definition of a real number? Some people see behind the beyond. Most people don't even see beyond the behind. Couldn't resist that old joke.  

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