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October 21st, 2018, 07:17 AM   #11
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 Originally Posted by v8archie 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.
With all due respect, but this post is just nonsense. I think you might want to study up on the hyperreal numbers before you make statements like these.

1) They most definitely are consistent.
2) The size of the infinitesimal term can depend on the approach taken to solve a problem? Not even sure what that means? You know there are a lot of infinitesimals in the hyperreals right, not just one.
3) "Whether or not hyperreals terms are consistent is immaterial to that." What??

 October 21st, 2018, 07:34 AM #12 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,557 Thanks: 2558 Math Focus: Mainly analysis and algebra With respect, I don't think you understand well enough what I'm saying to claim that it's nonsense. In particular, the last point that you take issue with seems very clear to me. I don't see how I can make it more so. I'll happily read about the consistency of the hyperreals if you can point me to something more than a throwaway comment on Wikipedia. But, like I say, that's not any great issue in the scheme of things.
 October 21st, 2018, 07:49 AM #13 Senior Member   Joined: Oct 2009 Posts: 688 Thanks: 223 Take any book on hyperreal numbers. For example, Goldblatt's "Lectures on the hyperreals". In his chapter 1-3 he gives an explicit construction of the hyperreal numbers in ZFC. Hence they are consistent (provided ZF theory is). I don't really know what you mean with "algebraically consistent" though. I tried googling it, but I didn't find anything. Maybe you could explain the term?
 October 21st, 2018, 08:02 AM #14 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,557 Thanks: 2558 Math Focus: Mainly analysis and algebra It means that using only logically accurate steps you could show that, for example $\epsilon = 0$. It something I understood from a comment by CRGreathouse a few years ago. I might be wrong, but like I say, it doesn't really matter.
October 21st, 2018, 08:10 AM   #15
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 Originally Posted by v8archie It means that using only logically accurate steps you could show that, for example $\epsilon = 0$. It something I understood from a comment by CRGreathouse a few years ago. I might be wrong, but like I say, it doesn't really matter.
But you can't do that. You construct the hyperreas inside ZFC, and inside ZFC you can prove that there are nonzero infinitesimals. So you can't prove it.

What is $\varepsilon$ anyway? Are you picking a specific infinitesimal?

October 21st, 2018, 10:31 AM   #16
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 Originally Posted by Micrm@ss What is $\varepsilon$ anyway? Are you picking a specific infinitesimal?
No. And I don't know the exact nature of the inconsistency (if it exists) either. That was just an example.

You can't prove the existence of non-zero infinitesimals, you have to assume that one exists, and by doing so you get them all.

October 21st, 2018, 10:37 AM   #17
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 Originally Posted by v8archie You can't prove the existence of non-zero infinitesimals
Of course you can. You construct them. I can give a very concrete example of infinitesimals in the hyperreals.

So the hyperreals are basically $$\mathbb{R}^\mathbb{N}$$ equipped with some equivalence relation. The equivalence relation is important in making a totally ordered field and should not be ignored, but I will ignore it anyway because it is unnecessary for the purposes here.

So every real is a hyperreal in the sense that every real $r$ corresponds to a hyperreal $(r,r,r,r,r,r,...)$.
An example of an infinitesimal would then just be $(1,1/2,1/3,1/4,....)$.

There are many other systems which allow infinitesimal vaues, such as the surreal numbers. Their construction there is equally simple.

October 21st, 2018, 10:43 AM   #18
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 Originally Posted by v8archie No. And I don't know the exact nature of the inconsistency (if it exists) either. That was just an example. You can't prove the existence of non-zero infinitesimals, you have to assume that one exists, and by doing so you get them all.
Archie you're embarrassing yourself. The Wiki articles on infinitesimals and the hyperreals are terrible. You'd do yourself a favor to Google and read some papers and articles on the hyperreals. Terence Tao's blog has a series of beautiful articles explaining ultrafilters and infinitesimals in a very comprehensible way. Tao's explanation of ultrafilters as voting systems is the most clear and insightful thing you'll ever read on this topic.

This goes for Aplanis also. Read these articles and stop expounding on things you don't understand. Forget Wiki, their articles on infinitesimals are simply awful, filled with inaccuracies.

For anyone interested in learning about the hyperreals, start with these three articles. Then Google around for any serious articles and papers about infinitesimals. Avoid Wiki like the plague.

https://terrytao.wordpress.com/2012/...dard-analysis/

https://terrytao.wordpress.com/2007/...on-management/

https://terrytao.wordpress.com/2013/...uous-analysis/

Quote:
 Originally Posted by AplanisTophet So I guess it depends on which set of hyperreal numbers you're talking about...
You get a different hyperreal field depending on the choice of ultrafilter. A striking theorem is that given CH, all the hyperreal fields are isomorphic.

https://math.stackexchange.com/quest...stem-ast-bbb-r

https://mathoverflow.net/questions/1...wer-of-mathbbn

ps Archie, You don't assume, you construct. You take the set of all sequences of reals, mod out by an equivalence relation, and voilà you get a field that contains infinitesimals. It's analagous to how you can construct the reals by taking the set of all sequence of rationals and modding out by Cauchy equivalence. The particular equivalence relation to get the hyperreeals is equality on some ultrafilter. So ultrafilters are where you need to start. They're initially confusing but ultimately very simple. Tao's articles are the best I've seen for demystifying the topic.

pps -- I never learned this in school. I was having one of those endless .999... arguments online a few years ago and as usual someone claimed the hyperreals falsify .999... = 1. (They don't. .999... = 1 is a theorem in the hyperreals). I spent a few weeks reading. Once I read Tao's articles it became much easier to read everything else. Point being that this material is accessible even to post-school adults. And sadly, it's simply not on Wikipedia. And much of what is on Wiki is misleading or outright wrong.

Last edited by Maschke; October 21st, 2018 at 11:11 AM.

October 21st, 2018, 11:05 AM   #19
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 Originally Posted by v8archie The hyperreals were constructed to provide solutions in the reals. Whether or not hyperreals terms are consistent is immaterial to that.

A lot of questions about real or even natural numbers require the use of the complex number system. Take for example the prime number theorem whose most elegant, simple proof requires complex numbers (there are proofs not requiring this of course, more about this later). There are also various integrals which can be computed trivially using the residue theorem.

Of course it is easy to see that every proof using the complex numbers can be modified into a proof not using them, but the proof will be substantially more complex (pun intended). In the same way as any proof using the hyperreals can be modified into a proof not using them. But such a proof might be more tedious.

Nowadays the complex numbers are an interesting field of study in their own right, but I think it is fairly safe to say that the complex numbers popped up as a method to proof stuff about the real numbers. The earliest such things were of course the soutions to cubic equations.

With that in mind, would you agree to the following modification of your statement:

Quote:
 Originally Posted by v8archie The complex numbers were constructed to provide solutions in the reals. Whether or not the complex numbers are consistent is immaterial to that.
Besides, do you believe the complex numbers to be consistent?

October 21st, 2018, 07:06 PM   #20
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 Originally Posted by Micrm@ss With that in mind, would you agree to the following modification of your statement
No. I understand that the complex numbers were "created" (defined might be better) so that all polynomials of degree $n$ would have $n$ roots.

Solving integrals and other uses came afterwards.

Quote:
 Originally Posted by Micrm@ss Besides, do you believe the complex numbers to be consistent?
Yes, but that's rather different in most contexts - we don't generally discard the imaginary part of complex numbers in producing our solutions, so consistency matters.

There may be other uses of infinitesimals where consistency does matter, where the infinitesimals aren't discarded. But I haven't heard of them.

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