I think Surreals and Hyperreals are serving a very similar function. The Surreals take a total order as a foundation, and then push that up to infinity, then define algebraic opperations. The Hyperreals start with the large set of indexed lists of real numbers, define algebraic opperations pointwise, and then to make them totally ordered. They both seem like they are about combining ordered set, and infinity, and getting the algebraic stuff for free.
Ordererd set is about < and >, which is what defines the Surreals/Hyperreals. Ordinals are not abot that, and are instead about pushing the notion of successor to infinity.
The problem for the Hyperreals for many applications is that aren’t uniquely ordered (sadly I can’t remember the name of the thing you need to define to get a unique ordering).
I think Surreals and Hyperreals are serving a very similar function. The Surreals take a total order as a foundation, and then push that up to infinity, then define algebraic opperations. The Hyperreals start with the large set of indexed lists of real numbers, define algebraic opperations pointwise, and then to make them totally ordered. They both seem like they are about combining ordered set, and infinity, and getting the algebraic stuff for free.
Ordererd set is about < and >, which is what defines the Surreals/Hyperreals. Ordinals are not abot that, and are instead about pushing the notion of successor to infinity.
The problem for the Hyperreals for many applications is that aren’t uniquely ordered (sadly I can’t remember the name of the thing you need to define to get a unique ordering).
Ultrafilter