A Class of Ethereal Borel Sets

We use the notation for ordinals as given in Section 1.6, where lowercase Greek letters such as $\alpha$, $\beta$, and $\omega$ denote ordinal numbers, $\prec$ and $\preceq$ denote the usual order relations among ordinals (with equality denoted by $\approx$), and for each ordinal $\alpha$, $P_{\alpha }=\left\{ \beta : 0 \preceq \beta \prec \alpha \right\}$ denotes the set of ordinal predecessors of $\alpha$. Here we let $\xi$ denote the first uncountable ordinal number (the ordinal called $\omega$ in Section 1.6's own notation, renamed here to $\xi$ to avoid clashing with this section's use of $\omega$ for a point of $(0,1]$).

For each $\alpha \in P_{\xi }$, find a sequence $\left\{ \beta_{\alpha }(n)\right\}_{n=1}^{\infty }$ of ordinals such that (i) $\beta_{\alpha }(n) \prec \alpha$ for all $n \in \mathbb {N}$ and (ii) the set $\left\{ \beta_{\alpha }(n): n \text{ is such that } \beta_{\alpha }(n)=\gamma \right\}$ is infinite for all $\gamma \prec \alpha$. Let $\mathcal{I}_{0}$ be as before, except the subscript $0$ is now the ordinal number $0$. For each $\alpha \in P_{\xi }$, with $\mathcal{I}_{\beta }$ defined for each $0 \preceq \beta \prec \alpha$, define $\mathcal{I}_{\alpha }=\left(\bigcup_{\beta \in P_{\alpha }} \mathcal{I}_{\beta }\right)^{*}$. This is a transfinitely inductive definition that produces the nondecreasing class $\left\{ \mathcal{I}_{\alpha }: \alpha \in P_{\xi }\right\}$. Next, given sets $A_{1}, A_{2}, \ldots$, define $\Psi \left(A_{1}, A_{2}, \ldots \right)=A_{1}^{c} \cup A_{2} \cup A_{3} \cup \cdots$ as before. Define $\Phi_{0}\left(A_{1}, A_{2}, \ldots \right)=A_{1}$. We define $\Phi_{\alpha }\left(A_{1}, A_{2}, \ldots \right)$ for all $\alpha \in P_{\xi }$ as follows: if $\Phi_{\beta }\left(A_{1}, A_{2}, \ldots \right)$ is defined for all $0 \preceq \beta \prec \alpha$, write

where $\left\{ m_{i j}: i, j \in \mathbb {N}\right\}$ is as given in this section.

Parts (a) and (b) correspond to Lemmas 1 and 2 of this section (Lemma 1: for $n \geq 0$ we can find $A_{1}, A_{2}, \ldots \in \mathcal{I}_{0}$ with $\Phi_{n}\left(A_{1}, A_{2}, \ldots \right)=(0,1]$, and we can find $A_{1}, A_{2}, \ldots \in \mathcal{I}_{0}$ with $\Phi_{n}\left(A_{1}, A_{2}, \ldots \right)=\varnothing$; Lemma 2: for $n \geq 0$ each $\mathcal{I}_{n}$-set has the form $\Phi_{n}\left(A_{1}, A_{2}, \ldots \right)$ for some appropriately chosen $A_{1}, A_{2}, \ldots \in \mathcal{I}_{0}$). No analog of Lemma 3 is needed. Part (c) corresponds to Lemma 5 (if $A_{1}, A_{2}, \ldots \in \mathscr {B}_{(0,1]}$, then $\Phi_{n}\left(A_{1}, A_{2}, \ldots \right) \in \mathscr {B}_{(0,1]}$ for all $n \geq 0$); no analog of Lemma 6 is needed. The notation preceding Lemma 7 and Lemma 7 itself (existence and uniqueness of the sequence $\left\{ \omega_{k}\right\}_{k=1}^{\infty }$ corresponding to $\omega \in (0,1]$) remain unchanged. We again let $I_{1}, I_{2}, \ldots$ denote a fixed listing of the countably many sets comprising $\mathcal{I}_{0}$. For each $\alpha \in P_{\xi }$, define the function $\varphi_{\alpha }:(0,1] \rightarrow 2^{(0,1]}$ by writing $\varphi_{\alpha }(\omega )=\Phi_{\alpha }\left(I_{\omega_{1}}, I_{\omega_{2}}, \ldots \right)$ whenever $\omega \leftrightarrow \left\{ \omega_{k}\right\}_{k=1}^{\infty }$. Next, for each $\alpha \in P_{\xi }$, let $B_{\alpha }$ denote the set $\left\{ \omega \in (0,1]: \omega \notin \varphi_{\alpha }(\omega )\right\}$. Part (d) is the analog of Lemma 8 (the set $B$ constructed in this section lies outside $\bigcup_{n=0}^{\infty } \mathcal{I}_{n}$). The sets $L_{k, n}$ and $D_{k}$ remain the same for all $k, n \in \mathbb {N}$; Lemmas 9, 10, and 11 (establishing $L_{k,n}, \left\{ \omega : \omega_{k}=n\right\}$, and $D_{k}$ are $\mathscr {B}_{(0,1]}$-sets) remain unchanged. Lemma 12 (the identity used to rewrite $\Phi_{n}$ of the $I_{\omega_{u_{k}}}$'s in terms of $\Phi_{n}$ of the $D_{u_{k}}$'s) is modified as in part (e). Finally, Lemma 13 ($B \in \mathscr {B}_{(0,1]}$) is modified to part (f).