October 22nd, 2018, 08:48 PM  #51  
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328  Quote:
But they are also an alternative model since they satisfy the same firstorder sentences as the reals. So according to firstorder logic, there is no difference between the reals and the hyperreals. Only secondorder sentences make the difference. In that point of view, they are alternative.  
October 23rd, 2018, 06:46 AM  #52 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
There seems to be a problem with the concept of infinitesimal in this thread. Perhaps I csn help: An infinitesimal is a distance which can be made arbitrarily small, as in 1/n for example. 
October 23rd, 2018, 07:41 AM  #53 
Senior Member Joined: Sep 2016 From: USA Posts: 645 Thanks: 408 Math Focus: Dynamical systems, analytic function theory, numerics  
October 23rd, 2018, 09:46 AM  #54 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
In keeping with the scope of this thread and forum, perhaps someone could illustrate the clarified and rigorous derivation of the derivatives of $\displaystyle x^{2}$ and sinx using hyperreals.

October 23rd, 2018, 10:29 AM  #55 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
Using your definition, it's the usual limit derivation.


Tags 
calculus, foundations, infinite, infinitesimal 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Infinitesimal  shaharhada  Calculus  8  October 20th, 2018 06:53 AM 
Infinitesimal  shaharhada  Algebra  9  November 10th, 2017 09:57 AM 
Calculus Foundations (Leibniz and Berkeley)  bt359  Calculus  1  November 15th, 2013 11:39 AM 
Infinite set mapping to infinitesimal interval  fjingram  Real Analysis  4  July 30th, 2010 04:07 AM 
help with the calculus of infinite sequence  1210  Real Analysis  3  September 20th, 2007 08:06 AM 