In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function $\Gamma: Y \rightarrow Z$ such that for all $y \in Y$, if $U_y \subseteq W$, then $\Gamma(y) \in W$ and $z \prec \Gamma(y)$ for all $z \in U_y$,

where $Y=2^{\omega \times \omega}$, $Z$ is the set of codes for linear orders on $\omega$ and $W=WO$.

But the reference [11] hadn't yet appeared and I'm able to find only the boundedness theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?

In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function $\Gamma: Y \rightarrow Z$ such that for all $y \in Y$, if $U_y \subseteq W$, then $\Gamma(y) \in W$ and $z \prec \Gamma(y)$ for all $z \in U_y$,

where $Y=2^{\omega \times \omega}$, $Z$ is the set of codes for linear orders on $\omega$, $W=WO$ and $U \subseteq Y \times Z$ is $Y$-universal for $\Sigma^1_1(Z)$.

But the reference [11] hadn't yet appeared and I'm able to find only the boundedness theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?

In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function \Gamma: Y \rightarrow Z such that for all y \in Y, if U_y \subseteq W, then \Gamma(y) \in W and z \prec \Gamma(y) for all z \in U_y,

where Y=2^(\omega \times \omega), Z is the set of codes for linear orders on \omega and W=WO.

But the reference [11] hadn't yet appeared and I'm able to find only the Boundedness Theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?

In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated:

Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a continuous function $\Gamma: Y \rightarrow Z$ such that for all $y \in Y$, if $U_y \subseteq W$, then $\Gamma(y) \in W$ and $z \prec \Gamma(y)$ for all $z \in U_y$,

where $Y=2^{\omega \times \omega}$, $Z$ is the set of codes for linear orders on $\omega$ and $W=WO$.

But the reference [11] hadn't yet appeared and I'm able to find only the boundedness theorem in [11], but not the uniform version. How can we show this theorem? Have you ever seen this uniform version somewhere?