Set of well-order codes ($\mathrm{WF}$)

The set $\mathrm{WF}\subseteq {\omega }^{\omega }$ consists of the reals coding well-orders of $\omega$. For $x\in \mathrm{WF}$, the norm $\Vert x\Vert$ is the order type of the well-order coded by $x$.

Every analytic set contained in $\mathrm{WF}$ has bounded rank: if $A\subseteq \mathrm{WF}$ is analytic, thenIn particular, the well-order codes produced continuously from all counterplays against one strategy have bounded ranks whenever they are all well-founded.

In the Solovay rank-comparison game, Players I and II produce $x,y\in {\omega }^{\omega }$; ill-founded codes lose before ranks are compared, and among well-order codes the prescribed inequality between $\Vert x\Vert$ and $\Vert y\Vert$ decides the winner. No strategy for Player I can uniformly produce a well-order code at least as long as every code produced by Player II.

There is a coordinatewise causal map $S:{\omega }^{\omega }\to {\omega }^{\omega }$ that recodes a relation after adjoining a new least element. If $x\in \mathrm{WF}$, then $S(x)\in \mathrm{WF}$ and $\Vert S(x)\Vert =\Vert x\Vert +1$; the tagged coding also ensures $S(x)\ne x$ when $x\notin \mathrm{WF}$. Because the first $n+1$ output coordinates depend only on the first $n+1$ input coordinates, Player II can produce $S(x)$ online.