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

The set of well-order codes $\mathrm{WF}\subseteq {\omega }^{\omega }$ consists of codes for strict well-orders $(D,R)$ on subsets $D\subseteq \omega$. The norm $\Vert x\Vert$ is their order type. Every countable ordinal, including finite ordinals and zero, has such a code.

For a coding convenient for games, ignore $x(0)$ as a tag. At coordinate $x(n+1)$ record the domain bit for $n$ and the finitely many relation bits $R(i,j)$ with $\max (i,j)=n$, in a fixed order. Read these bits from the binary expansion, ignoring unused higher bits. The code belongs to $\mathrm{WF}$ exactly when the decoded relation is a strict well-order of its decoded domain. This convention makes local recoding operations explicit and allows an arbitrary domain subset, rather than incorrectly requiring every well-order to have domain all of $\omega$.

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.

One explicit version of a Solovay rank-comparison game has Players I and II alternately choose natural numbers, producing $x,y\in {\omega }^{\omega }$. Declare that Player I wins exactly whenThus Player I must produce a valid well-order code whose rank is at least the rank of Player II's valid code. This payoff states the convention for invalid codes and for equality, which are part of the definition of the game.

Player II can win by producing the causal rank-raising map on well-order codes $S(x)$: a valid input gets rank $\Vert x\Vert +1$, and an invalid input already loses for Player I. Alternatively, if a strategy for Player I were winning, all its possible outputs would form an analytic set contained in $\mathrm{WF}$. The boundedness theorem for well-order codes bounds their ranks below a countable ordinal. Player II could simply play a code of a larger ordinal, contradicting that strategy. The game exposes how descriptive boundedness obstructs an online attempt to dominate every countable rank.

Using the coordinate convention in set of well-order codes, there is a causal map $S:{\omega }^{\omega }\to {\omega }^{\omega }$ that adds a new greatest element to the coded relation. Relabel every old element $n$ as $n+1$, adjoin $0$ to the domain, and place every old domain element below $0$. If $x\in \mathrm{WF}$, thenAdding a greatest element gives this successor rank; adding a least element would instead give $1+\Vert x\Vert$, which need not be larger for an infinite ordinal.

Set the unused tag $S(x)(0)=x(0)+1$, making $S(x)\ne x$ for every input. Coordinate1 records the new element0 and is constant. Coordinate $n+2$ records the relabelled old element $n+1$, its relations to earlier old elements, and its relation to the new greatest element; this uses only $x(n+1)$ and earlier input coordinates. Thus Player II can output $S(x)(m)$ immediately after seeing $x(m)$. Ill-foundedness of an old relation persists in its relabelled copy. This explicit coding justifies the online strategy in the Solovay rank-comparison game.