Scott Ranks of Classifications of the Admissibility Equivalence Relation

Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $\omega$. Let $\Phi : {}^\omega 2 \rightarrow S(\mathscr{L})$ be a $\Delta_1^1$ function with the property that for all $x,y \in {}^\omega 2$, $\omega_1^x = \omega_1^y$ if and only if $\Phi(x) \approx_{\mathscr{L}} \Phi(y)$. Then there is some $x \in {}^\omega 2$ so that $\mathrm{SR}(\Phi(x)) = \omega_1^x + 1$

The structural complexity of models of arithmetic

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than $\omega$. Other than that there are no restrictions. By giving a reduction via $\Delta^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $\alpha>\omega$ is realized as the Scott rank of a model of $T$

Trees of Scot rank ω\u3csub\u3e1\u3c/sub\u3e\u3csup\u3eCK\u3c/sub\u3e, and computable approximability

Makkai [10] produced an arithmetical structure of Scott rank ω 1 CK . In [9], Makkai\u27s example is made computable. Here we show that there are computable trees of Scott rank ω 1 CK . We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω 1 CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind

Classification from a computable viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory. Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings. Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work

The countable admissible ordinal equivalence relation

Let F_(ω1) be the countable admissible ordinal equivalence relation defined on ^ω2 by x F_(ω1) y if and only if ω_1^x=ω_1^y. Some invariant descriptive set theoretic properties of F_(ω1) will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed F_(ω1) is not the orbit equivalence relation of a continuous action of a Polish group on ^ω2. Becker stengthened this to show F_(ω1) is not even the orbit equivalence relation of a Δ_1^1 action of a Polish group. However, Montalbán has shown that F_(ω1) is Δ_1^1 reducible to an orbit equivalence relation of a Polish group action, in fact, F_(ω1) is classifiable by countable structures. It will be shown here that F_(ω1) must be classified by structures of high Scott rank. Let E_(ω1) denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E ≤ aΔ_1^1 F denotes the existence of a Δ_1^1 function f:X→Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies E_(ω1) ≤ aΔ_1^1 F_(ω1). However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), E_(ω1) ≤ aΔ_1^1 F_(ω1) is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Δ_1^1 reducible to F_(ω1). This shows the consistency of a negative answer to a question of Sy-David Friedman