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I have heard this stuff called non-standard analysis. It introduces hyper reals $-$ an extension to real numbers $-$ to deal with infinitesimals. Now if you extend the real number line, how do you extend it? Shall we create another coordinate on $y$ axis (if our real number line is $x$ axis) OR shall we create a whole new infinitesimal number line at the point zero on real number line like: enter image description here

The latter seems intuitive but is there any problem in imagining it that way?

Joe
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  • The latter is a more precise image. Just note though that an infinitesimal is not zero - the new axis should be labelled with $\varepsilon\cdot 0,\varepsilon\cdot 1,\dots$. – Wojowu Mar 25 '17 at 13:19
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    @Wojowu ... except the zero, that is considered an infinitesimal. –  Mar 25 '17 at 13:20
  • @Masacroso From what I have heard, an infinitesimal is a positive number smaller than all positive reals, so $0$ doesn't qualify as such. – Wojowu Mar 25 '17 at 13:23
  • @Wojowu It is more complex than this. I read two books (not completely, just some chapters each one) some time ago about the matter I was shocked by this, but it is how it is in the theory. But to be honest I dont remember too much... There are various ways to define, formally, non-standard analysis, what I read was based in the internal set theory background, that is an extension of ZFC. In general non-standard analysis is a lot more complex than the standard one... despite it is based in the "intuitive idea" of infinitesimal. –  Mar 25 '17 at 13:26
  • @Wojowu: If infinitesimal is not zero, then on which point in the real number line we are creating a whole new infinitesimal number line? – Joe Mar 25 '17 at 13:33
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    @Wojowu If your goal is to redevelop simple real analysis imo the most motivated definition is a number whose standard part is 0 (thus including both 0 and negatives). This opinion coming mostly from having done so myself and trying out various definitions (although I have references that use the same convention). – GPhys Mar 25 '17 at 13:36
  • Or is imagining a whole new infinitesimal number line on a point in the real number line wrong? – Joe Mar 25 '17 at 13:36
  • @GPhys: Do you mean the real value of infinitesimal is zero even though hyper real value is non zero? If this is right it seems to me there is nothing wrong in imagining a whole new infinitesimal number line at point zero as said in the question. Anything wrong? – Joe Mar 25 '17 at 13:41
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    @Masacroso I am aware of all the subtelties regarding the definitions of nonstandard analysis, but in any fixed such system saying "bigger than zero and smaller than all positive reals" does make sense. On the second thought though, I do have to agree with GPhys that taking his definition of an infinitesimal does seem to be the most natural one. I can't remember where, but I swear I have seen a definition somewhere in which positivity was required. – Wojowu Mar 25 '17 at 13:42
  • @Wojowu This is mostly because I almost always wanted to make general statements about infinitesimals in the above sense and only very rarely wanted (or needed) a positive infinitesimal (when introducing Riemann integration). – GPhys Mar 25 '17 at 13:42
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    In the hyperreal number line, every real number (not just zero) is surrounded by a “halo” of hyperreal numbers which lie infinitesimally close to it. But the hyperreal line is not just “denser”, it's also “longer”, since there are also numbers greater than any real number, such as $\omega=1/\epsilon$ if $\epsilon$ is a positive infinitesimal. – Hans Lundmark Mar 25 '17 at 15:29
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    @Joe Try first chapter of https://www.math.wisc.edu/~keisler/calc.html –  Mar 26 '17 at 15:16

2 Answers2

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By the transfer principle, anything you do with the standard reals looks exactly the same when done internally to the hyperreals.

In particular, there is a hyperreal number line. And (internally) it looks exactly like the standard number line.

The value of nonstandard analysis comes in comparing the real and hyperreal number lines. When overlaid atop one another, you'll find:

  • Every standard real number is also a hyperreal number
  • Every standard real number has a halo of hyperreal numbers surrounding it, and this halo doesn't contain any standard reals.
  • There are no "gaps"; if $a<x<b$ where $a,b$ are standard real and $x$ is hyperreal, then $x$ is in the halo of some standard real number
  • There are more hyperreals off to the right and left, larger in magnitude than anything in the halo of a standard real.

If you take the picture of the extended reals (and extended hyperreals) instead — that is, add $\pm \infty$ as the endpoints of the number line, then the hyperreals of the last bullet can be gathered up into the halos of $+\infty$ and $-\infty$.

(note, still, that every hyperreal is still smaller than $+\infty$; it's just that they're 'closer' to $+\infty$ than any standard real, or even any hyperreal in the halo of a standard real)

So, the picture you are trying to imagine looks fairly reasonable. Keisler's book uses something like that a lot, where you look at the standard number line, and then when desired, you use a "telescope" to "zoom in" on some point to see an infinitesimal segment of the hyperreal line.

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The correct/usual way to think about it is the following number line \begin{equation*} [\, \underbrace{\cdots\quad-\nu-1\quad\phantom{+}-\nu\phantom{+\,}\quad-\nu+1\quad\cdots}_{\text{negative "infinite" hyperreals}} \,]\,\cdots\,[\, \underbrace{\cdots\,\,\,-1\quad0\quad1\quad \cdots}_{\text{usual reals}} \,]\,\cdots\,[\, \underbrace{\cdots\quad\nu-1\quad\phantom{+}\nu\phantom{+\,}\quad\nu+1\quad\cdots}_{\text{positive "infinite" hyperreals}} \,] \end{equation*} with the additional note that every point $c$ on the above number line is surrounded by a family of points infinitesimally close to it: \begin{equation*} \cdots\quad c-2\varepsilon\quad c-\varepsilon\quad\phantom{+}c\phantom{+\,}\quad c+\varepsilon\quad c+2\varepsilon\quad\cdots \end{equation*}

GPhys
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