Going beyond the surreal numbers

Question

Denote the class of surreal numbers No. We can create new "number", like the gap $\infty=\{\infty^L|\infty^R\}$, defined by $\infty^L=\{x:\exists n\in\mathbb N,x<n\}$ and $\infty^R=\{x:\forall n\in\mathbb N,x>n\}$. $\infty\notin$No because $\infty^L$ and $\infty^R$ are not sets, but classes. Apparently we can construct new numbers in the form $\{L|R\}$, where $L$ and $R$ are subclasses of No such that no $x\in L$ is $\geq$ any $y\in R$. Then we can use these new numbers to construct more numbers, and continue to expand No (possible indefinitely?). Is there any reason we do not do so?

Answer 1

Relentlessly filling cuts is of course the main construction idea of the surreal numbers---at every ordinal birthday, one fills all the cuts that exist in the previously-born surreals. Your proposal is to continue filling cuts after all ordinal birthdays are completed.

All such cuts will have cofinality Ord on one side or the other, and so each such cut will take a proper class to represent it. So the first thing to say is that there will be certain set-theoretic foundational difficulties with undertaking the construction. For example, this is not straight-forwardly a ZFC construction, but you could proceed in GBC for a step or so. To proceed much further, you will need stronger second-order set-theoretic axioms, such as the axiom ETR of elementary transfinite recursion, which allows one to undertake recursions on proper class well-founded relations whose rank exceeds Ord.

But another perspective is that what you are proposing is just the situation that arises when one has a smaller universe $V_\kappa$ extended to the full universe $V$. The surreal numbers $\text{No}^{V_\kappa}$ as constructed up to $\kappa$ are the surreal numbers of the universe $V_\kappa$, which proceeded in $\kappa$ many birthdays, but the surreal numbers of the full universe $V$ simply carried on with birthday $\kappa$, filling the cuts that you describe, and then $\kappa+1$ and so on through the ordinals of $V$.

One can imagine a hierarchy of universes $$V_\kappa\prec V_\lambda\prec V_\theta\prec\cdots$$ and the surreal numbers of each of them are related to the next in just this way.

So your construction is realized by considering the surreal numbers as constructed in various universes and realizing how each next universe continues the cut-filling construction of the surreals of the smaller universes.