Conway's left and right "cuts" look like sets but then again I'm not really sure what they are. Is this alien mathematics accidentally discovered on Earth by perhaps the most intelligent Earthling who ever lived. I don't know. All I can say is I took a detour from my usual math studies to look into Surreal numbers and it was an interesting trip. I'm quite conservative when it comes to math. I don't go around preaching Surreal numbers are the way to do math but it is a curiosity none-the-less worth at least the small price of reading Knuth's short book about it from cover to cover.
As formal as it gets.
Edit: (not intending to be harsh, just that I probably lack some context to properly understand what you are asking).
Just not sure about your point.