October 22nd, 2018, 09:48 PM  #51  
Senior Member Joined: Oct 2009 Posts: 687 Thanks: 223  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, 07:46 AM  #52 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,628 Thanks: 119 
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, 08:41 AM  #53 
Senior Member Joined: Sep 2016 From: USA Posts: 534 Thanks: 306 Math Focus: Dynamical systems, analytic function theory, numerics  
October 23rd, 2018, 10:46 AM  #54 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,628 Thanks: 119 
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, 11:29 AM  #55 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,557 Thanks: 2558 Math Focus: Mainly analysis and algebra 
Using your definition, it's the usual limit derivation.


Tags 
calculus, foundations, infinite, infinitesimal 
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