October 20th, 2018, 09:38 AM  #1  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Infinitesimal and Infinite and Foundations of Calculus Quote:
Example: Divide a line into n seqments. As n becomes infinite the distance between points becomes infinitesimal. Basis of integral. n never equals infinity, you can never get there, and segment is never zero, two points define a segment Last edited by skipjack; October 20th, 2018 at 10:33 AM.  
October 20th, 2018, 10:55 AM  #2 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187 
I really don't get it, zylo. You say: "n never equal infinity" in your third sentence. Ok, I kind of understand this from your point of view. But then your second sentence is like "As n becomes infinite". Doesn't n equal infinity then?? If n can never equal infinity, what does the sentence "As n becomes infinite the distance between points becomes infinitesimal." mean then? Last edited by skipjack; October 20th, 2018 at 11:42 AM. 
October 20th, 2018, 12:15 PM  #3  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra  Quote:
It's not the basis of the derivative or the integral at all. The hyperreal system makes use of infinitesimals, but then ignores them in producing answers for the real system. Calculus in the reals is the method that doesn't require such approximation, producing solutions without recourse to infinitesimals.  
October 20th, 2018, 12:19 PM  #4 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187  What do you mean with that? The definition of limit, derivative and integral in the hyperreal number system depends very crucially on the infinitesimal. Why do you say it is being ignored? Just because the final answer is real?

October 20th, 2018, 04:12 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
To return real results you discard any infinitesimal parts of the answer. It's been demonstrated that this always produces valid answers in the reals, but it's no better than the original concerns with Newton/Leibnitz work when viewed in the context of the hyperreals. The hyperreals are a tool for working in the reals, they aren't intended to be a completely consistent system in their own right. 
October 20th, 2018, 05:39 PM  #6  
Senior Member Joined: May 2016 From: USA Posts: 1,192 Thanks: 489  Quote:
 
October 20th, 2018, 10:17 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
Well I read the text book with varying degrees of thoroughness.

October 21st, 2018, 03:50 AM  #8  
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187  Quote:
Or are you saying that Robinson did not intend them to be a completely consistent system in their own right? Both statements are very wrong in my opinion. But I'm not sure which of the two (if any) you are claiming.  
October 21st, 2018, 05:35 AM  #9  
Senior Member Joined: Jun 2014 From: USA Posts: 413 Thanks: 26  Quote:
Quote:
https://en.wikipedia.org/wiki/Hyperreal_number The same article states that the use of the definite article 'the' in the phrase 'the hyperreal numbers' is "somewhat misleading in that there is not a unique ordered field that is referred to in most treatments." So I guess it depends on which set of hyperreal numbers you're talking about...  
October 21st, 2018, 07:10 AM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra  Quote:
I believe that the proof Aplanis Tophet refers to is that the results obtained for the reals by using hyperreals are consistent if and only if the results obtained without are consistent. The hyperreals were constructed to provide solutions in the reals. Whether or not hyperreals terms are consistent is immaterial to that.  

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calculus, foundations, infinite, infinitesimal 
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