Balogh finite restriction data

Using the ordinary ordered-pair coding, functions on $\kappa$ with binary values and the finite tuples and power sets displayed here have ranks below $\kappa +\omega$: each pairing or power set increases the rank by finitely many levels, and the function domains consist of ordinals below $\kappa$. Thus they lie in $V_{\theta }$ by F2. Apply F1 and A1 with the finite parameter set $\{\kappa ,\omega ,f,g,h\}$ to obtain countable $M$. By F2–F3, the ranks of its countably many elements have supremum below $\theta$, and adding one still stays below the limit cardinal $\theta$. Hence $M\in V_{\theta }$. A second application of F1 with $M$ as a parameter gives countable $N$ containing it. Enumerations of countable subsets of either model, and finite sets of elements of these models, also have bounded rank below $\theta$, by F2–F3. We have used a set structure throughout.

For either elementary substructure $P$, the empty set and each next finite ordinal are uniquely definable from the preceding one, so all natural numbers lie in $P$. If a function and an argument belong to $P$, its uniquely specified value belongs to $P$ by elementarity. The same uniqueness argument for finite set formation shows every finite subset of $P$ belongs to $P$. If $D\in P$ is externally countable and nonempty, an actual enumeration $\omega \to D$ belongs to $V_{\theta }$ by the rank bound in step 1.1. Elementarity supplies such an enumeration in $P$. Its values at every natural number are in $P$, so $D\subseteq P$; the empty case holds directly. The assertion that the enumeration is a function onto $D$ is absolute here: all quantifiers in its definition are bounded by its graph, $\omega$, or $D$, whose elements are in the transitive $V_{\theta }$ by F2. In particular $M\in N$ and countability give $M\subseteq N$.

If distinct $c,c^{\mathrm{\prime }}\in C\cap N$ are given, some $\xi <\kappa$ has $c(\xi )\mathrm{\ne }{c}^{\mathrm{\prime }}(\xi )$. This is a bounded assertion about their graphs and $\kappa$ and hence holds in $V_{\theta }$ by F2. Elementarity supplies such a $\xi \in N$, which belongs to $A$. Their restrictions to $A$ differ, proving injectivity on $C\cap N$. If $c\upharpoonright A\in B$, there is $c^{\mathrm{\prime }}\in C\cap M$ with that restriction; step 2.1 puts $c^{\mathrm{\prime }}$ in $N$, so injectivity gives $c=c^{\mathrm{\prime }}\in M$. Conversely membership in $C\cap M$ gives that trace in $B$ by the definition. This proves both directions of the stated trace test.

Since $M\subseteq N$, one has $a\subseteq A$. For $\alpha \in A$, the values $g(\alpha )$ and $h(\alpha )$ belong to $N$ by step 2.1, because their functions belong to $M\subseteq N$. They are finite sets, so that same step puts every member in $N$. Thus the traces in $q(\alpha )$ come from $C\cap N$, and $h(\alpha )\subseteq A$. These establish the displayed types, and evaluation $v_{\alpha }$ has domain exactly the finite $q(\alpha )$. To count root triples, fix an enumeration of countable $B$; finite subsets are encoded by finite sets of natural-number indices, using binary sums, and a function to $2$ on such a finite set has finitely many codes. Pairing these codes with $n<\omega$ gives a countable set of triples. $I_T$ is a subset of it, and each required $K_T(t)$ has a nonempty witness set by the definition of $I_T$; A1 selects them, including the empty selection when $I_T=\mathrm{\varnothing }$. All defining types and checks follow. QED.