Some new computable structures of high rank

Self Cite

We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: Every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show the complexities are maintained in the sense that if the functor is ∆ 0 α , then the interpretation that induces it is ∆ in α up to ∆ 0 α equivalence. Group in Logic and the Methodology

Section: ) (Where Pmentioning

confidence: 99%

Borel Functors and Infinitary Interpretations

Harrison-Trainor¹,

Miller²,

Montalbán³

2018

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“…For some time the most important open question about structures of high Scott rank was whether there is a computable structure of Scott rank CK 1 whose computable infinitary theory is not ℵ 0 -categorical-which, by the corollary above, is exactly the same as asking for a computable structure of Scott complexity Π CK 1 +1 . Eventually Harrison-Trainor, Igusa, and Knight [7] provided such an example.…”

Section: The Standard Examples Of Computable Structures Of Scott Rank Ckmentioning

confidence: 99%

“…Proof. It is known that there exists a computable structure C of Scott complexity Π +1 ([13] for the case < CK 1 , and [7] for the case = CK 1 ). Let A be a computable structure of Scott complexity Σ +1 .…”

Section: A Has a D -σ α Scott Sentence If And Only If For Some C ∈ A (A C) Has A π αmentioning

confidence: 99%

Scott Complexity of Countable Structures

Alvir¹,

Greenberg²,

Harrison-Trainor³

et al. 2021

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We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$ , $\Pi _{\alpha }$ , or $\mathrm {d-}\Sigma _{\alpha }$ . We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$ . There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$ , $\Pi _{\omega _1^{CK}+1}$ , $\Sigma _{\omega _1^{CK}+1}$ . Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$ . The existence of such structures was an open question.

“…Calvert et al [5] and Knight and Miller [12] found examples of computable structures with Scott rank CK 1 . Recently, Harrison-Trainor et al [8] gave examples of structures of Scott rank CK 1 whose computable infinitary theory is not ℵ 0 -categorical.…”

mentioning

confidence: 99%

The Complexity of Scott Sentences of Scattered Linear Orders

Alvir¹,

Rossegger²

2020

We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank we show that it has a Scott sentence. It follows from results of Ash [2] that for every countable there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal or Scott sentence and give a characterization of those linear orders of rank with optimal Scott sentences. At last we show that for all countable the class of Hausdorff rank linear orders is complete and obtain analogous results for index sets of computable linear orders.