October 21st, 2018, 06:17 AM  #11  
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323  Quote:
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, 06:34 AM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 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, 06:49 AM  #13 
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323 
Take any book on hyperreal numbers. For example, Goldblatt's "Lectures on the hyperreals". In his chapter 13 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, 07:02 AM  #14 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 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, 07:10 AM  #15  
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323  Quote:
What is $\varepsilon$ anyway? Are you picking a specific infinitesimal?  
October 21st, 2018, 09:31 AM  #16  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra  Quote:
You can't prove the existence of nonzero infinitesimals, you have to assume that one exists, and by doing so you get them all.  
October 21st, 2018, 09:37 AM  #17 
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323  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, 09:43 AM  #18  
Senior Member Joined: Aug 2012 Posts: 2,343 Thanks: 732  Quote:
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/...dardanalysis/ https://terrytao.wordpress.com/2007/...onmanagement/ https://terrytao.wordpress.com/2013/...uousanalysis/ Quote:
https://math.stackexchange.com/quest...stemastbbbr https://mathoverflow.net/questions/1...werofmathbbn 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 postschool 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 10:11 AM.  
October 21st, 2018, 10:05 AM  #19  
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323  Quote:
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: Besides, do you believe the complex numbers to be consistent?  
October 21st, 2018, 06:06 PM  #20  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra  Quote:
Solving integrals and other uses came afterwards. 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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calculus, foundations, infinite, infinitesimal 
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