Imagining hyper reals in non-standard analysis

Question

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:

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

Answer 1

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*}

Answer 2

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.