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The surreal numbers $\mathbb{SN}$ form a class of numbers introduced by J.H. Conway, which behave as an ordered field (even if technically it is not a set). In particular, Conway showed that every ordered field can be embedded by $\mathbb{SN}$ as it is isomorphic to a subfield of this class.

For each ordinal $\alpha$, it is possible to define a subfield $\mathbb{SN}_{\alpha}$ of $\mathbb{SN}$ such that: (i) $\mathbb{SN}_{\alpha} \subset \mathbb{SN}_{\beta}$ for every $\alpha < \beta$, (ii) the 'union' of the sets $\mathbb{SN}_{\alpha}$ over each ordinal $\alpha$ equals the class $\mathbb{SN}$. Using this definition, we may assign to each ordered field $\mathbb{F}$ an order type $ot(\mathbb{F})$ equal to the smallest ordinal $\alpha$ such that $\mathbb{F} \leq \mathbb{SN}_{\alpha}$.

Given two ordinals $\alpha / \rho / \pi$, we may define an $(\alpha,\rho,\pi)$-chain as a chain of ordered fields $(\mathbb{F_{\beta}})_{0 \leq \beta \leq \rho}$ such that:

(i) $ot(\mathbb{F}_0) = \alpha, ot(\mathbb{F}_{\rho}) = \pi$,

(ii) for each ordinal $\beta$, $\mathbb{F}_{\beta+1}$ is a one-element extension of $\mathbb{F}_{\beta}$,

(iii) for each limit ordinal $\beta$, $\mathbb{F}_{\beta} = \cup_{\alpha < \beta} \mathbb{F}_{\alpha}$.

We may then define $F(\alpha,\rho)$ as the largest $\pi$ such that there exists an $(\alpha,\rho,\pi)$-chain, and $G(\alpha)$ as the smallest $\beta$ such that $F(\alpha,\beta) = \beta$. It seems that this definition could provide an interesting "functor" on ordinals, although the following is not clear to me: (i) is this notion robust wrt to the choice of ordinal sequences, (ii) how does it relate with known functors such as Veblen functions?

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    $\begingroup$ The class of surreal numbers is usually denoted No rather than $\mathbb{SN}$. Also, the properties you have mentioned on $\mathbb{SN}_\alpha$ do not determine them uniquely, and by making them much bigger (while still satisfying your properties), the value of $F(\alpha,\rho)$ will obviously be affected. In particular, since you have no continuity assumption on $\mathbb{SN}_\alpha$, isn't it clear that $F$ will not be so nice? Perhaps you mean something more specific than what you have said about $\mathbb{SN}_\alpha$, such as the field generated by all surreals born before day $\alpha$? $\endgroup$ Apr 22, 2014 at 12:49

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