Three Dollar Quill

Harrumph. I'm still recovering, but I at least was able to take some notes on arithmetic primitives. I've got a sketch of an idea for going from addition and fixed rotation, to negation, and subtraction, which lays the groundwork for multiplication. Complex conjugate is over to the side. I trust these ideas, some more than others, but I at least want an inkling of how the equivalence classes required for these to be numbers are going to work, before I fully commit.

Here's what I'm seeing:

• All triples have access to three sets.
• Those sets can be empty.
• A triple that has one non-empty set containing a single number, I'm pretty sure is going to be a number.
• I believe it's possible for a triple containing just two non-empty sets, each containing a single number, to not be a number.
• Therefore, I need some kind of binary relation that can be adjusted in some way based on the indices of the sets being compared.
• The equivalent relation on surreal numbers is "greater than or equal to" or "not greater than nor equal to"; this is distinct from "less than".
• I guess in some sense, that makes the nimbers "like NaN", except you can do math to them and it sometimes cancels to a number.
• However, trying to natively support the complex plane means that ordering is not well-defined.
• If indices influence the relation, that seems to help.
• For example, "morally", {|ω*|ω} should be a number (-1), while {|ω|ω*} should not, and wow my text editor is highlighting that wrong.
• So, perhaps if there are specific projections onto the reals, or something like that, then the ordering relation can be recovered.
• Not sure if there's a natural interpretation of such a thing, or if it's just like "look it's easy, just multiply both sides by one of these constants, then add each result to its own complex conjugate and go".
• Actually, that can't be right, because it assumes that we're working with proper numbers, which is exactly the thing we want to prove...
• Except that if we're working on the elements of the sets, then the set elements are simpler, so we can use structural induction...

Well, I've put myself in an awful position for it, but I just realized that, as far as time goes, what I should be doing now is getting to bed.

Good night.