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October 22nd, 2018, 08:48 PM   #51
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Quote:
 Originally Posted by v8archie So they are, as I always thought, an expansion of the reals. They aren't even really an alternative model, they are built out of reals to get more, as the reals are built from the rationals.
Yes, they are an expansion of the reals, so you do get more.

But they are also an alternative model since they satisfy the same first-order sentences as the reals. So according to first-order logic, there is no difference between the reals and the hyperreals. Only second-order 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: 124 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
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Quote:
 Originally Posted by zylo 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.
lmao. Thank for clearing that up!

 October 23rd, 2018, 09:46 AM #54 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 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: 7,616 Thanks: 2605 Math Focus: Mainly analysis and algebra Using your definition, it's the usual limit derivation.

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