The oscillation block lemma

Fix Skolem functions for a sufficiently large $H(\mathrm{\Theta })$. The cuts $M\cap {\omega }_1$ of countable elementary Skolem hulls containing the fixed parameters form a club $C$: larger countable ordinal parameter sets give unboundedly many cuts, and unions of increasing $\omega$-chains give closure. Intersect $C$ with the club supplied by F1. By F3 and F7 this intersection meets the stationary set $\{\delta :w_{\delta }=w\}$. Choose $M$ whose cut $\delta =M\cap {\omega }_1$ is such a point. This is the sole model-selection use of AC.

Choose initial $a\in A$ and $b\in B$ with every coordinate strictly above $\delta$; only finitely many members of either pairwise-disjoint family can contain $\delta$. Apply the strict branch of F1 to this pair and call its outputs $a_0,b_0$. The appended nonempty block is the final part of every $L(\delta ,b_0(j))$, and F1 makes the comparison strict on that block. Hence $e_{a_0(i)}(\max L(\delta ,b_0(j)))>e_{b_0(j)}(\max L(\delta ,b_0(j)))$ for all $i,j$.

Suppose $a_m,b_m$ have been chosen, the previously marked points lie below both first-disagreement bounds to the next stage, and the last point of $L(\delta ,b_m(j))$ has $e_{a_m(i)}>e_{b_m(j)}$ for every $i,j$. Apply [F1] first with equality, producing a common nonempty block $K_m^{=}$ on which the new $e$-values agree, and then to that output with strict inequality, producing a common nonempty block $K_m^{>}$ on which the next $a$-values exceed the next $b$-values. Put ${\xi }_m=\min K_m^{>}$. The two trace splices give $L(\delta ,b_m(j))<K_m^{=}<K_m^{>}$, independently of $j$, and both new minima have label $w_{\delta }=w$.

On the old trace, the first-disagreement bounds preserve every comparison. At the first point of $K_m^{=}$ the previous comparison is strict $>$ and the new comparison is equality, so [F2] creates no downward crossing. Comparisons remain equality through that block. At ${\xi }_m$, equality at its predecessor changes to strict $>$, so exactly ${\xi }_m$ is added to every coordinatewise oscillation set; the comparison stays strict through $K_m^{>}$, so no other new crossing appears. Label restriction in [F1] preserves all old marked labels and assigns $w$ to ${\xi }_m$.

Finite induction therefore produces sequences $a_m,b_m$ above $\delta$ and increasing ${\xi }_m$ with the seven invariants used in Moore's proof: proper common trace extension, one new oscillation, preservation below the first-disagreement bounds, a terminal strict comparison, old-label restriction, and label $w$ at every marked point.

Fix $n<\omega$. By [F4], choose ${\gamma }_0<\delta$ above every $L(\delta ,b_n(j))$ and every disagreement below $\delta$ between $e_{b_m(j)}$ and $e_{b_{m^{\mathrm{\prime }}}(j)}$ for $m,m^{\mathrm{\prime }}\le n$. Put ${\gamma }_1=\max {\bigcup }_{j<l}L(\delta ,b_n(j))$. For each $i<k$, the restriction $r_i=e_{a_n(i)}\upharpoonright ({\gamma }_1+1)$ belongs to $M$: the corresponding restriction of $e_{{\gamma }_1+1}$ belongs to $M$, and [F4] says that $r_i$ is a finite modification of it. Hence $S=\{a\in A:(\mathrm{\forall }i<k)e_{a(i)}\upharpoonright ({\gamma }_1+1)=r_i\}$ belongs to $M$. It contains $a_n\mathrm{\notin }M$, so it is uncountable; if it were countable, an enumeration in $M$ would put every member of $S$ in $M$.

By the limit law in [F5], choose $\eta <\delta$ such that $\eta <\xi <\delta$ implies ${\gamma }_0<\min L(\xi ,\delta )$. Since $S$ is an uncountable pairwise-disjoint family and $\eta$ is countable, some member of $S$ has all coordinates above $\eta$; this assertion has parameters in $M$, so elementarity gives such an $a\in S\cap M$. Thus $a<\delta$, ${\gamma }_0<L(a(i),\delta )$, and the definition of $S$ gives $L(\delta ,b_n(j))<\mathrm{\Delta }(e_{a(i)},e_{a_n(i)})$ for every $i,j$. Since every $b_m$ lies above $\delta$, one also has $a<b_m$ for all $m\le n$.

The inequalities in step 7.1 let [F5] splice every trace through $\delta$, and [F3] gives the corresponding labelled splice $\mu (a(i),b_m(j))=\mu (a(i),\delta )\cup \mu (\delta ,b_m(j))$. On $L(a(i),\delta )$ the functions $e_{b_m(j)}$ do not depend on $m$, by the choice of ${\gamma }_0$; on $L(\delta ,b_m(j))$, step 5.1 added exactly the marked oscillations and preserved their labels. The terminal comparison on the first piece is strict, so the splice boundary creates no further $\le$-to-$>$ oscillation. Hence clauses 2--4 hold, while step 7.1 gives clause 1. For $n=0$ the same reflection chooses $a$ and all unions over marked points are empty.

Since the recursively chosen sequence $⟨b_m:m<\omega ⟩$ is fixed before $n$ is specified, step 8.1 proves the required quantifier order. No coordinate assignment was introduced: the conclusion holds for all $i<k,j<l$ simultaneously.