If you add +1 up from 0 and do −1 from w you never cross because those are part of separate archimedean fields. I think there is also a result that if you have surreals and add the limitation that they need to be archimedean you get the real numbers
Real number have the completeness axiom/property. While for lesser systems the limit fails to exist and thus it is missing a “completing piece” for surreals the limit is not unique. Between the ascending numbers and what would be the real limit there is quaranteed to be a surreal number, so in that sense they are “overcomplete” to follow the standard limit constructions. That is {series|}=U is always going to exist but so will {series|U}=V while for reals V is required to either equal U or equal one of the series.
Surreals are more rich than hyperreals. One could imagine a dart board where some areas scored 10 points, some lines scored 15 points and intersections of lines score 20 points. Treating anything that isn’t area as having literally zero measure the disintions between lines being more probable than line intersections get lost (0=0). With surreals one could try to say that llines are infidesimal in respects to areas and intersection points are infinidesimal with respect to lines. And you could still hold that twice area is twice as likely and twice the line length is twice as likely and twice the amount of points is twice as likely. But areas, lines and points would seem to live in 3 disconnected lands. An abundance of points would not be able to make them reach line probability (unless you provide an actual infinite amount)
However if intersections scored infinite points in respect to lines then it could make expected value sense to aim for intersection points. But if we know that all points are not infinidesimal in respect of each other then we know that any intersection point aiming strategy is inferior.
I guess the connection here is that if you try to divide the dart board into small areas, they are still going to be areas and some mechanism might misvalue an area containing an intersection point that whole area is worth what the point is worth.
I intend not to or atleast the sense is meant to be compatible with the mathematical word. Looking at wikipedia article for “Field” subsection “ordered fields” there is the mention that real are the unique complete field up to isomorphism and there are multiple subsections in surreals that could contend to be isomorphic. Real multiples of w also have an additive inverse and multiplicative inverse making them “another copy” of the reals. I think I might have thougt in error that the other subsection would fullfill field axioms in its own right but rather it is just a subsection that can be mapped to reals.
If you add +1 up from 0 and do −1 from w you never cross because those are part of separate archimedean fields. I think there is also a result that if you have surreals and add the limitation that they need to be archimedean you get the real numbers
Real number have the completeness axiom/property. While for lesser systems the limit fails to exist and thus it is missing a “completing piece” for surreals the limit is not unique. Between the ascending numbers and what would be the real limit there is quaranteed to be a surreal number, so in that sense they are “overcomplete” to follow the standard limit constructions. That is {series|}=U is always going to exist but so will {series|U}=V while for reals V is required to either equal U or equal one of the series.
Surreals are more rich than hyperreals. One could imagine a dart board where some areas scored 10 points, some lines scored 15 points and intersections of lines score 20 points. Treating anything that isn’t area as having literally zero measure the disintions between lines being more probable than line intersections get lost (0=0). With surreals one could try to say that llines are infidesimal in respects to areas and intersection points are infinidesimal with respect to lines. And you could still hold that twice area is twice as likely and twice the line length is twice as likely and twice the amount of points is twice as likely. But areas, lines and points would seem to live in 3 disconnected lands. An abundance of points would not be able to make them reach line probability (unless you provide an actual infinite amount)
However if intersections scored infinite points in respect to lines then it could make expected value sense to aim for intersection points. But if we know that all points are not infinidesimal in respect of each other then we know that any intersection point aiming strategy is inferior.
I guess the connection here is that if you try to divide the dart board into small areas, they are still going to be areas and some mechanism might misvalue an area containing an intersection point that whole area is worth what the point is worth.
Ah, you are using “field” in a different sense (than “something with addition and multiplication obeying the usual laws”).
I intend not to or atleast the sense is meant to be compatible with the mathematical word. Looking at wikipedia article for “Field” subsection “ordered fields” there is the mention that real are the unique complete field up to isomorphism and there are multiple subsections in surreals that could contend to be isomorphic. Real multiples of w also have an additive inverse and multiplicative inverse making them “another copy” of the reals. I think I might have thougt in error that the other subsection would fullfill field axioms in its own right but rather it is just a subsection that can be mapped to reals.