Scott spectral gaps for trees are bounded

Abstract: Given a Borel class of trees, we show that there is a tree in that class whose Scott sentence is not too much more complicated than the definition of the class. In particular, if the class is definable by a $Π_α$ sentence, then there is a model of Scott rank at most $α+ 2$. This gives another proof-and one that does not require first proving Vaught's conjecture for trees-of the fact that trees are not faithfully Borel complete.

• The paper proves that every satisfiable iubscriptlphand-subscript sentence describing rooted trees has a model with a iubscriptlpha+3nd-subscript Scott sentence and Scott rank at most lpha+2nd-subscript, improving the analogous bound for linear orders.
• The authors combine the ubscriptlphand-subscript/ubscriptlphand-subscript infinitary hierarchies with a Henkin-style forcing construction to build generic trees whose automorphism orbits have low-complexity definitions.
• A iubscript[?2nd-subscript theory of trees is exhibited for which every model has Scott rank at least 3, while the bounded-gap theorem independently shows that trees are not faithfully Borel complete.],
• follow_up_questions:[

Overview and main results

This paper, by Harrison-Trainor and Kim, establishes that Scott spectral gaps are bounded for the class of (rooted, connected, acyclic) trees in the language $\mathcal{L}_{tree} = \{r, P\}$, where $r$ names the root and $P$ is the parent relation. The central result is: given a satisfiable $\Pi_\alpha$ sentence $\varphi$ of trees, there is a tree $A \models \varphi$ admitting a $\Pi_{\alpha+3}$ Scott sentence, hence of Scott rank at most $\alpha + 2$, where Scott rank is taken in Montalbán's robust sense. This is a direct analogue—and an improvement—of the earlier result for linear orders due to Gonzalez and Harrison-Trainor, which gave a bound of $\alpha + 4$. The improvement arises because trees admit a decomposition into disjoint descendant subtrees, eliminating the need for the Lindenbaum–Tarski-style argument about overlapping initial segments that was required for linear orders.

The paper also proves a non-trivial lower bound: there exists a $\Pi_2$ theory $r$0 of trees such that no model of $r$1 has a $r$2 Scott sentence, so every model has Scott rank at least $r$3. As with linear orders, this leaves an open gap between the lower bounds and the upper bounds on the possible Scott ranks within a theory's spectrum.

Motivation: faithful Borel completeness

The principal motivation comes from the theory of faithful Borel embeddings. Steel proved Vaught's conjecture for trees (in a more general sense of "tree" than used here), while Friedman–Stanley showed trees are Borel complete. Neither result transfers to all countable structures because, as Gao showed, trees are not faithfully Borel complete: the saturation of the image of a Borel reduction from graphs to trees cannot be Borel. Harrison-Trainor and Gonzalez previously established that bounded Scott spectral gaps imply failure of faithful Borel completeness, via an argument that does not require first proving Vaught's conjecture for the class—unlike Gao's approach, which went through a strengthening of Vaught's conjecture for linear orders and trees. The present theorem therefore yields a second proof that trees are not faithfully Borel complete, independent of any progress on Vaught's conjecture for trees.

The authors note that Boolean algebras are Borel complete but it is open whether they are faithfully Borel complete; establishing bounded Scott spectral gaps for Boolean algebras would be a route to resolving that question without first resolving Vaught's conjecture there.

Technical apparatus

The proof relies on two hierarchies of $r$4 formulas beyond the standard $r$5 hierarchy. Karp's theorem characterizes asymmetric back-and-forth relations $r$6 in terms of $r$7/\,$r$8 truth-preservation. However, the relation $r$9 is not itself $P$0-definable; instead the paper uses the $P$1/$P$2 hierarchies of Chen, Gonzalez, and Harrison-Trainor, where $P$3 closes lower-complexity formulas under existential quantification and countable disjunction, and dually for $P$4. Two facts are essential: the Karp-type characterization holds verbatim with $P$5/$P$6 in place of $P$7/$P$8, and for every tuple there exists an $P$9 formula defining exactly those tuples above it in the $\Pi_\alpha$0 ordering. These hierarchies matter because a formula of the shape $\Pi_\alpha$1 is only $\Pi_\alpha$2 despite being $\Pi_\alpha$3—a strictness that drives the improved bounds.

The tree-theoretic analogue of interval decomposition for linear orders is the descendant tree $\Pi_\alpha$4: the subtree rooted at $\Pi_\alpha$5 consisting of its descendants avoiding the tuple $\Pi_\alpha$6. This is $\Pi_\alpha$7-definable by a formula $\Pi_\alpha$8, so relativization preserves syntactic complexity. The key structural lemma states that $\Pi_\alpha$9 holds if and only if $\varphi$0 and each corresponding descendant tree satisfies the same relation. A companion syntactic lemma shows that if a tuple satisfies an $\varphi$1 formula, one can extract $\varphi$2 sentences describing the descendant trees that suffice to recover the formula. Notably, these lemmas require tuples to be closed under ancestors; the paper justifies this convention with a lemma showing that closure under predecessors does not change the back-and-forth relations. The syntactic splitting lemma also holds only for $\varphi$3, not $\varphi$4—the counterexample being precisely the lower-bound construction of the final section.

The forcing construction

Fixing an $\varphi$5 sentence $\varphi$6 (sufficient since each $\varphi$7 sentence is $\varphi$8), the authors build a countable existential fragment $\varphi$9 of $A \models \varphi$0 formulas containing $A \models \varphi$1, closed under relativization to descendant trees, together with a countable stock $A \models \varphi$2 of witness trees realizing each satisfiable fragment formula and supplying the descendant-tree sentences from the syntactic lemma.

Sentences in $A \models \varphi$3 (the satisfiable ones) either force unity (any two extensions remain jointly satisfiable) or force splitting (every extension has two incompatible subextensions). A tree is generic if every finite subtree satisfies some forcing sentence and every satisfiable fragment formula is either realized or explicitly contradicted relative to each root-containing finite subtree. A tree has property $A \models \varphi$4 if any two same-level nodes either satisfy incompatible sentences or a common unity-forcing sentence.

The main existence theorem constructs a generic tree with property $A \models \varphi$5 satisfying $A \models \varphi$6 via a Henkin-type argument over countably many constants. Each stage maintains a finite satisfiable set of $A \models \varphi$7 sentences witnessed by a tree in $A \models \varphi$8. The steps handle disjunction witnesses, Henkin constants for existentials (introducing ancestor constants as needed), conjunct and universal instantiation schedules, atomic diagram completion, and the forcing conditions. The critical step toward property $A \models \varphi$9 performs "surgery": given constants $\Pi_{\alpha+3}$0 with $\Pi_{\alpha+3}$1 describing the descendant trees below $\Pi_{\alpha+3}$2, if incompatible extensions $\Pi_{\alpha+3}$3, $\Pi_{\alpha+3}$4 both remain consistent, they are added separately; otherwise $\Pi_{\alpha+3}$5 is shown to be satisfiable and to force unity, and is imposed on both nodes simultaneously. Satisfaction is preserved by grafting a model of the new sentence onto the relevant node of the witnessing tree.

Verification: orbits and Scott sentences

The verification proceeds through two lemmas. First, if a unity-forcing sentence holds below $\Pi_{\alpha+3}$6 in a generic tree, then every extension below it also satisfies unity-forcing sentences at each coordinate—genericity propagates the condition downward. Second, defining a Scott-like sentence as one that forces unity and forces all descendant trees below any finite tuple to do likewise, any two generic trees satisfying the same Scott-like sentence on corresponding descendant trees are isomorphic, via an explicit back-and-forth family built from matching unity-forcing sentences.

The concluding observation is that every automorphism orbit of a generic tree with property $\Pi_{\alpha+3}$7 is $\Pi_{\alpha+3}$8-definable. For a tuple $\Pi_{\alpha+3}$9, each coordinate receives either its Scott-like sentence or, in the splitting case, an $\alpha + 2$0 formula built by existentially quantifying over a splitting configuration and conjoining negations of the sentences distinguishing all same-level alternatives—a construction enabled directly by property $\alpha + 2$1. The conjunction with the quantifier-free tree-isomorphism type gives an $\alpha + 2$2 definition of the orbit; injectivity follows by induction up the tree using the back-and-forth lemma. Since having all automorphism orbits $\alpha + 2$3-definable is equivalent to possessing a $\alpha + 2$4 Scott sentence (a strengthening, in the direction used here, of Montalbán's characterization via $\alpha + 2$5-definability), the constructed model has Scott rank at most $\alpha + 2$6, yielding the stated $\alpha + 2$7/$\alpha + 2$8 bounds for the original $\alpha + 2$9 sentence.

Lower bound

The lower-bound example works with coloured trees: height-one trees whose children are distinguished by infinite binary colour codes, subject to the requirements that distinct children differ in some colour and every finite colour pattern is realized. This $\alpha + 4$0 theory is effectively bi-interpretable with an uncoloured tree construction, transferring the result to the pure tree language. Every automorphism orbit is $\alpha + 4$1-definable by the full colour pattern, giving a $\alpha + 4$2 Scott sentence; but no orbit of a non-root element can be $\alpha + 4$3-definable even with parameters, since any finitary quantifier-free formula sees only finitely many colours and another node agreeing on those but differing elsewhere would satisfy the same formula without being automorphic. By Montalbán's orbit-definability characterization, no model has a $\alpha + 4$4 Scott sentence.

Limitations and open questions

The upper and lower bounds do not meet: the paper determines that some $\alpha + 4$5 theories force all models to have Scott rank at least $\alpha + 4$6, while guaranteeing a model of rank at most $\alpha + 4$7 (for $\alpha + 4$8, the bound gives rank at most $\alpha + 4$9 after translation). Whether the true spectral gap for trees is smaller than $\Pi_2$0, or whether the lower bound can be pushed higher, remains open. More broadly, the method depends on the disjointness of descendant trees—the analogue of Remark on the Lindenbaum–Tarski product argument makes explicit that no overlapping-decomposition issue arises for trees—and whether analogous bounded-gap results hold for classes lacking such clean decompositions, most prominently Boolean algebras, is unresolved and constitutes the natural next target given the connection to faithful Borel completeness discussed in the introduction.

Conclusion

The paper proves that every $\Pi_2$1 theory of trees has a model of Scott rank at most $\Pi_2$2, improving the known bound for linear orders and providing a new, self-contained proof that trees fail to be faithfully Borel complete—one that bypasses any appeal to Vaught's conjecture. The argument combines the $\Pi_2$3/$\Pi_2$4 hierarchy with a Henkin forcing construction producing generic trees whose orbits are uniformly low-level definable, exploiting the disjoint decomposition of trees into descendant subtrees. Together with the $\Pi_2$5 lower-bound example, the work sharpens the picture of Scott spectra for trees while leaving the precise gap between bounds open, and pointing toward Boolean algebras as the next class where bounded spectral gaps would settle an outstanding question about faithful Borel completeness.