Borel Rank Research Articles

A descriptive Main Gap Theorem

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text].

THE WADGE ORDER ON THE SCOTT DOMAIN IS NOT A WELL-QUASI-ORDER

AbstractWe prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets$\mathbb{P}_{emb} $equipped with the order induced by homomorphisms is embedded into the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. We then show that$\mathbb{P}_{emb} $admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain.

Locally finite ω‐languages and effective analytic sets have the same topological complexity

Local sentences and the formal languages they define were introduced by Ressayre in . We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the set of Borel ranks of locally finite ω‐languages is the ordinal , which is strictly greater than the first non‐recursive ordinal . This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet and also by Duparc, Finkel, and Ressayre in . Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system . Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable.

Higher Randomness and Forcing with Closed Sets

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259---297, 1975) that there exists a largest ź11${\Pi ^{1}_{1}}$ set of measure 0. An explicit construction of this largest ź11${\Pi ^{1}_{1}}$ nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495---508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131---1148, 2015) and Yu (Fundam. Math. 215, 219---231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely Σ30${\Sigma }^{0}_{3}$. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Σ30${\Sigma }^{0}_{3}$. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

Wadge Degrees of Infinitary Rational Relations

We show that, from the topological point of view, 2-tape Buchi automata have the same accepting power as Turing machines equipped with a Buchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape Buchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time Buchi 1-counter automata or by Buchi Turing machines. In particular, for every non-null recursive ordinal $\alpha$, there exist some $\Sigma^0_\alpha$-complete and some $\Pi^0_\alpha$-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal $\gamma^1_2$ which is strictly greater than the first non-recursive ordinal $\omega_1^{CK}$. This very surprising result gives answers to questions of Simonnet (1992) and of Lescow and Thomas (1988,1994).

Graev metric groups and Polishable subgroups

We prove a conjecture of Hjorth: There is an uncountable Polish group all of whose abelian subgroups are discrete. We first construct directly a witness to Hjorth's conjecture. Then we consider an existing example in the literature. The example is the metric completion of a free topological group constructed by Graev. We give a definition slightly more general than Graev's and prove some properties of the Graev metrics which seem to be unknown previously. We also consider the problem of finding Polishable subgroups of the Graev metric groups with arbitrarily high Borel rank. In doing this we prove some general theorems on extensions of Polish groups with this property.

Borel ranks and Wadge degrees of context free $\omega$-languages

We show that the Borel hierarchy of the class of context free $\omega$-languages, or even of the class of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Borel hierarchy of the class of $\omega$-languages accepted by Turing machines with a Buchi acceptance condition. In particular, for each recursive non-null ordinal $\alpha$, there exist some ${\bf \Sigma}^0_\alpha$-complete and some ${\bf \Pi}^0_\alpha$-complete $\omega$-languages accepted by Buchi 1-counter automata. And the supremum of the set of Borel ranks of context free $\omega$-languages is an ordinal $\gamma_2^1$ that is strictly greater than the first non-recursive ordinal $\omega_1^{\mathrm{CK}}$. We then extend this result, proving that the Wadge hierarchy of context free $\omega$-languages, or even of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Wadge hierarchy of $\omega$-languages accepted by Turing machines with a Buchi or a Muller acceptance condition.

Sharper changes in topologies

Let G be a Polish group, ia Polish topology on a space X, G acting continuously on (X, ir), with B C X G-invariant and in the Borel algebra generated by ir. Then there is a larger Polish topology -r* D ir on X so that B is open with respect to r*, G still acts continuously on (X, -r*), and -r* has a basis consisting of sets that are of the same Borel rank as B relative to -r.

Louveau's theorem for the descriptive set theory of internal sets

AbstractWe give positive answers to two open questions from [15]. (1) For every set C countably determined over , if C is then it must be over , and (2) every Borel subset of the product of two internal sets X and Y all of whose vertical sections are can be represented as an intersection (union) of Borel sets with vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when X is a measurable space and Y a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy.

Polish group actions and the Vaught conjecture

We consider the topological Vaught conjecture: If a Polish group $G$ acts continuously on a Polish space $S$, then $S$ has either countably many or perfectly many orbits. We show 1. The conjecture is true for Abelian groups. 2. The conjecture is true whenever $G$, $S$ are recursively presented, the action of $G$ is recursive and, for $x \in S$ the orbit of $x$ is of Borel multiplicative rank $\leq \omega _1^x$. Assertion $1$ holds also for analytic $S$. Specializing $G$ to a closed subgroup of $\omega !$, we prove that nonempty invariant Borel sets, not having perfectly many orbits, have orbits of about the same Borel rank. An upper bound is derived for the Borel rank of orbits when the set of orbits is finite.

Projective subsets of separable metric spaces

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A ⊆ X is Σ 1 1( X) iff there exists a complete separable metric space Y and Borel set B ⊆ X × Y such that A = { x ϵ X : ∃ y ϵ Y ( x, y) ϵ B}. Except for the fact that X may not be completely metrizable, this is the classical definition of analytic set and hence has many equivalent definitions, for example, A is Σ 1 1(X) iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by Σ X 1 or abstract projective subset of X. A set of A ⊆ X is Σ X 1 iff there exists an n ϵ ω and a Borel set B ⊆ X × X n such that A = { x ϵ X:∃ y ϵ X n ( x, y) ϵ B}. These sets ca n be far more pathological. While the family of sets Σ 1 1( X) is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where Σ X 1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ 1 1( X) set is Borel in X but there exists a Σ 1 1( X 2) set which is not Borel in X 2. The space X 2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ 1 1( X) and Σ X 1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam.

Bounds on Scott rank for various nonelementary classes

Ideally we would like to answer the following question. Suppose T is a first order theory and there is a bound ~<0~1 on the Scott rank of countable models of T. What can we say about 7? In I-S] Sacks points out that ~ < 6~(T) and asks if this can be improved to ~ < co~. If T has only countably many countable models, then Martin 's conjecture would imply ~ < ~o. 2. While this question seems very difficult, it is possible to obtain exact bounds on Scott rank in some more abstract settings. For example in [ K M S ] we showed that i fE is S~ equivalence relation where every class is Borel and there is a bound on the Borel rank of the equivalence classes, then ~ < 7,1 (we define 7z 1 below) and 7~ is the optimal bound. In this paper we will consider bounds on Scott rank for PCL~I -sentences. Basically we will show that 62~(~o) is an optimal bound on the Scott ranks of countable models a PCL~ -sentence ~o and 7~0P) is an optimal bound on the well founded models of an L,ol`o-sentence ~p. These results are proved in Sect. 2 and Sect. 3. In Sect. 4 we examine bounds on Scott rank for these classes when we also consider uncountable models. In this case the bounds coincide. The results of Sect. 4 were obtained earlier in unpublished work by Nadel and Stavi.