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arXiv · 1702.06587

There is no classification of the decidably presentable structures

Matthew Harrison-Trainor
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Abstract

A computable structure $\mathcal{A}$ is decidable if, given a formula $\varphi(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $\varphi$ holds of $\bar{a}$. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is $\Sigma^1_1$-complete. This result holds even if we restrict out attention to groups, graphs, or fields. We also show that the index sets of the computable structures with $n$-decidable presentations is $\Sigma^1_1$-complete for any $n$.

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Matthew Harrison-Trainor. 2017-02-21. There is no classification of the decidably presentable structures. https://arxiv.org/abs/1702.06587

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