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October 20th, 2018, 09:38 AM   #1
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Infinitesimal and Infinite and Foundations of Calculus

Quote:
 Originally Posted by Micrm@ss I'm all for keeping zylo on this forum and discussing with him in HIS threads.
INFINITESIMAL: Arbitrarily small distance between two points which can never equal zero. Basis of derivative.

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: 905 Thanks: 351 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
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Quote:
 Originally Posted by zylo INFINITESIMAL: Arbitrarily small distance between two points which can never equal zero. Basis of derivative. Example: Divide a line into n seqments. As n becomes infinite the distance between points becomes infinitesimal. Basis of integral.
This is wrong. An arbitrarily small number is still finite - such as $\epsilon$ in the definition of limits. Just as an arbitrarily large number is finite.

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
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Quote:
 Originally Posted by v8archie The hyperreal system makes use of infinitesimals, but then ignores them in producing answers for the real system.
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: 7,696 Thanks: 2681 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
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Quote:
 Originally Posted by v8archie 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.
Do you have any citations proving that, or is that your opinion?

 October 20th, 2018, 10:17 PM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra Well I read the text book with varying degrees of thoroughness.
October 21st, 2018, 03:50 AM   #8
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Quote:
 Originally Posted by v8archie 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.
Are you saying that they are not a completely consistent system in their own right?

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
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Quote:
 Originally Posted by v8archie The hyperreals are a tool for working in the reals, they aren't intended to be a completely consistent system in their own right.
Quote:
 Originally Posted by Micrm@ss Are you saying that they are not a completely consistent system in their own right? 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.
According to Wiki, Robinson showed the hyperreals were consistent if and only if the reals were. "This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated."

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
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Quote:
 Originally Posted by Micrm@ss Are you saying that they are not a completely consistent system in their own right?
I'm led to believe that they are not algebraically consistent, which I take to mean that the size of the infinitesimal term can depend on the approach taken to solve a problem.

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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