Surreals have multiples and are ordered, yet they contain multiple different archimedian fields. You can have for all r in reals and all s in surreals that r*s exists and that there is another surreal that is greater than all of r*s. Arbitrarily large finite is a different thing than an infinitely large value. You can’t “inch” your way to infinity. If you have a single bad experience and “inch” around it you will only reach one archimedian field but how do you know that you have covered the whole space of bad experience?
Surreals have multiples and are ordered, yet they contain multiple different archimedian fields. You can have for all r in reals and all s in surreals that r*s exists and that there is another surreal that is greater than all of r*s. Arbitrarily large finite is a different thing than an infinitely large value. You can’t “inch” your way to infinity. If you have a single bad experience and “inch” around it you will only reach one archimedian field but how do you know that you have covered the whole space of bad experience?