He would have a point... — SophistiCat

Ha! Saw what you did there...

So if I line up all the surreal numbers along a line, and we ignore the imaginaries for a bit, what number is missing?

My understanding is, none.

Perhaps until someone builds some more numbers.

Whatever line the surreal is on, I'm still wondering how you would describe writing down the decimal expansion of it.

My guess is that the real number requires 2^N = ω digits, and when you've got that, then to approach a surreal, you just add whole lot more digits. That's a guess, but I suspect it doesn't work. But what else would there be? It seems unreasonable to say there exists no system of signs in which they could be represented. And this obviates the need for a two or more number lines.

Or another way; I've seen Conway's notation for surreals, e.g., {4 l 5} = 4 1/2. And how he extends it for irrationals. But how does his method of extension uniquely yield surreals?

Perhaps until someone builds some more numbers. — Banno

As I understand it, it is somehow provable that there cannot be more numbers than the surreals.

Yes, that's what I understand, too.

But one ought not underestimate the power of human creativity...

↪fishfry
Whatever line the surreal is on, I'm still wondering how you would describe writing down the decimal expansion of it. — tim wood

An infinitesimal is technically a "mathematical quantity", but not a real number. Real numbers have decimal expansions. Infinitesimals have an arithmetic that is not the same as real numbers. r+r=r, e.g. How would that work with a hypothetical decimal expansion? Not all arithmetic is with real numbers.

I tend to look at physics to see what kinds of math are consistent with the physical world. Virtually all I see there is real and complex analysis, functional analysis, group theory, matrix theory, etc. None of which seem related to anything but real or complex numbers. (Well, the Hahn-Banach theorem in FA generates lots of functionals and requires in its proof a single transfinite step, unless this is avoided by requiring a tad more in the hypotheses) . Even string theory - seen as a flop by many - doesn't invoke esoteric number systems.

But I read that surreals are connected to game theory. So what do I know? :roll:

How would that work with a hypothetical decimal expansion? — jgill

I'm still wondering how you would describe writing down the decimal expansion of it. — tim wood

There's a decimal-like notation for the hyperreals, called the Lightstone notation.

https://en.wikipedia.org/wiki/A._H._Lightstone

The Wiki article on the Surreals doesn't mention anything about notation, so it's likely that there is no notation yet discovered for them. The article mentioned that the Surreals contain all the ordinals; and there aren't workable notations for all the ordinals as far as I know. I'm under the impression that there aren't even notations for all the countable ordinals but I'm fuzzy on this. Clues might be found here:

https://en.wikipedia.org/wiki/Ordinal_notation

There's a decimal-like notation for the hyperreals, called the Lightstone notation — fishfry

I keep learning things on this forum. I wonder if hyperreals will ever supplant the real number system. Infinitesimal calculus, touted as a more intuitive way to teach the subject, shows up here and there around the world, even in some high schools. The following seems to be a kind of advanced calculus course centered on the hyperreals that complements a similar elementary calculus course that apparently was abandoned years ago:

https://www.math.wisc.edu/~keisler/foundations.pdf

The author even has diagrams showing "where" infinitesimals and transfinites are located on the real line! :cool:

Those diagrams make me think of a way of visualizing transfinite numbers I’d thought of before, which I realize now could also be used to visualize infinite’s number.

The transfinite visualization is to imagine the real number line projected sort of logarithmically, so that on the left side you have zero and one the normal distance away from each other, but then the numbers get closer and closer the further right you go until at some finite distance right they “reach infinity”; then you put omega there, omega plus one a single unit right of that, and then repeat the whole logarithmic acceleration until twice omega is twice as far right as omega, then repeat that again. Possibly take that whole new transfinite number line and project it logarithmically the same way to reach even bigger transfinite numbers even faster.

For the infinitesimals, do the same thing, except each “omega” is instead a real number, and the logarithmically projected numbers that asymptotically approach each real number are infinitesimals.

Could be. I'm awaiting an application of the hyperreals that is useful in describing or predicting physical phenomena. Perhaps quantum theory will be couched in those terms at some point. But for now they seem to be pretty darn abstract.