The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and calculus inside it. This completion contains all its gaps as new numbers. A few examples of gaps are $On = \{No |\}$ which is the gap at the end of the number line, $\infty = \{\mathbb{N}|{x \in No: \forall y \in \mathbb{N} (x>y)}\}$ which is the gap between all finite numbers and all infinite numbers, $1/On = \{0|No^+\}$ which is the gap between zero and all positive numbers and $1/\infty = \{{x \in No: \forall y \in \mathbb{R}^+ (x<y)}|\mathbb{R}^+\}$ which is the gap between all infinitesimals and all positive reals. These new "numbers" don't form a field however, as we lose associativity for example (eg. since $On + On = On$ then $(On + On) - On \neq On + (On - On)$).
From a different angle, when studying combinatorial game theory (which as practically how surreal numbers got invented) one can stumble upon the concept of a "loopy game". A loopy game is a game where one of the players have the option to move to the current position, effectively skipping their turn. For example $A = \{A|B\}$ is a loopy game, as it has itself as one of its options. These loopy games violate well foundedness. Loopy games can also be included in the theory of the surreal numbers if we reject the axiom of foundation. However, like the gaps, arithmetic with them is weird. For example if we take $on = \{on|\}$ (which is a very large game, larger than all surreal numbers) then again $on + 1 = on$. But we can keep associativity by sacrificing the ability to have additive (and multiplicative it seems) inverses. So for example $on - on$ does not necessarily equal $0$.
Now it seems to me that there is a very clear connection - or at least a correspondence - between gaps and loopy games. Each gap can be identified with a loopy game. The obvious example is $On = \{No |\}$ and $on = \{on|\}$. They are both larger than all surreal numbers, and they both have similar arithmetic. Another example: $\infty = \{\mathbb{N}|{x \in No: \forall y \in \mathbb{N} (x>y)}\}$ can be identified with the loopy game $\infty = \{\mathbb{N}|\infty\}$ which are both larger than all finite numbers but smaller than all infinite numbers. And $1/On = \{0|No^+\}$ can be identified with the loopy game $over = \{0|over\}$ which are both smaller than every positive surreal.
I've also noticed that we can effectively get this by permitting the proper classes used to define gaps to be (inconsistent) sets. For example, the gap $On$ uses the proper class of all surreal numbers. If this were to be a set, it would contain $On$ since its left and right classes are now both sets, making it a surreal number. Thus $On$ would become an element of its own left set (thus becoming a loopy number) and, since it is clearly the biggest number in that set, it would make it equal to $\{On|\}$, exactly equivalent to the $on = \{on|\}$ we saw before. Same thing happens with every other gap. As another example, the right set of $\infty$ is a proper class. Making it a set causes $\infty$ to becoming a surreal number, and since it is bigger than all finite numbers, that makes it an element of its own right set! And again, since it is the smallest element in that set, that makes it equal to $\{\mathbb{N}|\infty\}$.
How I imagine this all works is that the gaps sort of locate where there are missing numbers and the corresponding loopy numbers "fill the gaps". You cannot create anymore numbers because any attempt to do so will result in a number already created. For example trying to find a number between $over$ and $0$ by taking $\{0|over\}$ simply lands you back at $over$. So, unlike with the gaps, where the limitation to make anymore numbers was placed artificially and axiomatically (since we can't have a proper class be a member of another class by definition), the loopy games really do fill all the gaps and make sure that there are none left. Including the loopy numbers in the class of surreal numbers gives an "absolutely continuous" collection similar to that of the Dedekind completion of the surreal sumbers. Why is the Dedekind completion preferred over this and does anyone know of any studies done on this? Is there something deeper going on with the proper class/gap $\iff$ non-well-founded set/loopy game correspondence?
