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arXiv · 1712.00847

Scott Ranks of Classifications of the Admissibility Equivalence Relation

William Chan·Matthew Harrison-Trainor·Andrew Marks
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Abstract

Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $ω$. Let $Φ: {}^ω2 \rightarrow S(\mathscr{L})$ be a $Δ_1^1$ function with the property that for all $x,y \in {}^ω2$, $ω_1^x = ω_1^y$ if and only if $Φ(x) \approx_{\mathscr{L}} Φ(y)$. Then there is some $x \in {}^ω2$ so that $\mathrm{SR}(Φ(x)) = ω_1^x + 1$.

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William Chan, Matthew Harrison-Trainor, Andrew Marks. 2017-12-03. Scott Ranks of Classifications of the Admissibility Equivalence Relation. https://arxiv.org/abs/1712.00847

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