Understanding how to construct a tall narrow tree

Question

I am trying to read the following presentation by Hamkins: http://jdh.hamkins.org/wp-content/uploads/2017/01/Bonn-Logic-Seminar-2017.pdf

At page 30(34) there is a lemma called "Uniform covering implies approximation" which I have a hard time understanding.

The claim is the following:

If $W\subseteq V$ has $\lambda$-uniform $\delta$-cover property, with $\delta$ regular and $\lambda$ strong limit, then it has the $\delta^+$-approximation property for subsets of $\lambda$.

Here is the proof given in the presentation:

Proof. Assume $s\in2^{\lambda}$ has all $\delta^{+}$-small approximations in $W$; aim to show $s\in W$. Assume inductively that $s\restriction_{\alpha}\in W$ for all $\alpha<\eta$. By uniform covering, can find a tree $T\in W$ height $\eta$, levels size $<\delta$, such that $s\restriction_{\alpha}\in T$ for $\alpha<\eta$. If $\delta<cof\left(\eta\right)$, this is tall narrow tree, and so $s\in W$. Otherwise, $cof\left(\eta\right)\leq\delta$. By a simple closure argument, find cofinal $J\subseteq\eta$ size $\delta$, such that distinct nodes $p,q$ on level $\beta\in J$ have $p\left(\alpha\right)\ne q\left(\alpha\right)$ some $\alpha\in J$. By approximation assumption, $s\restriction_{J}\in W$, and this determines $s\restriction_{\eta}$.

I don't understand how is the tree in the first part defined, I find it confusing since $\eta$ might be greater then $\delta$.

If needed I can add the definitions from the presentation.

Answer 1

What a nice coincidence, I was actually present at that talk!

We want to show the following:

There is a tree $T\subseteq {}^{<\eta}2$ in $W$ (ordered by end-extension) with levels of size ${<}\delta$ so that for any $\alpha<\eta$, $\left.s\right|_\alpha\in T_\alpha$.

Observe that the problem at hand is only interesting when $\delta\leq\eta<\lambda$.

First note that $\lambda$ is a strong limit cardinal in $W$, too. Thus there is some $\theta<\lambda$ and a bijection $h:({}^{<\eta}2)^W\rightarrow \theta$ in $W$.

We now make use of the third variant of the $\lambda$-uniform $\delta$-covering as defined in the slides:

Whenever $f:\lambda\rightarrow\lambda$ is in $V$, there is $B\subseteq\lambda\times\lambda$ in $W$ with all vertical slices of size ${<}\delta$ and $f\subseteq B$.

As $\theta,\eta<\lambda$, this surely still holds if we let $f$ be a function $f:\eta\rightarrow\theta$ and demand $B$ to be a subset of $\eta\times\theta$. In our case, we take $$f:\eta\rightarrow\theta,\ \alpha\mapsto h(\left.s\right|_\alpha)$$ This is well-defined by our assumtion that $\left.s\right|_\alpha\in W$ for all $\alpha<\eta$. Hence we get a $B\subseteq \eta\times\theta$ in $W$ with vertical silces of size ${<\delta}$ and $f\subseteq B$. We can furthermore assume that whenever $(\alpha,\beta)\in B$, then $h^{-1}(\beta)\in{}^\alpha 2$. You can think of the vertical slice of $B$ at $\alpha<\eta$, that is $B_\alpha=\{\beta\mid (\alpha, \beta)\in B\}$, (or rather of $h^{-1}[B_\alpha]$) as a guess for $\left.s\right|_\alpha$. This readily gives us the desired tree $T$ in $W$: The $\alpha$-th level of $T$ is $$T_\alpha=\{t\in {}^\alpha 2\mid h(t)\in B_\alpha\}$$