I would like to know a good way of sketching the surreal number line. I wonder if there is a format that is widely used.
In this video, Conway makes a quick drawing of a line, just like the real line (without gaps), and separates the integers to the infinities with some visual space, but no particular notation. However, I started wondering what would be a good way to draw the infinitesimals in the line. I do not think it is possible to do the same thing he did for infinities in this case, because it may give the impression that a real number has a "next" number.
For example:
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___ 0_$\ldots\epsilon\ldots$_? ____1 _2 _3 ___$\ldots$ __$\omega-1$ _$\omega$ ___
I wonder if representing the infinitesimals, or gaps (like in page 37 ONAG2) for that matter, in a precise manner is at all possible. Would a skecth like the one found above, put together with an explanation of its limitations, be a valid representation?
There are more surreal numbers than real ones, so they can't each get their own point on a line. In fact, the surreal numbers comprise a proper class, i.e. no set, be it of the points on a line or otherwise, is as big as the class of surreal numbers.