The theory of hereditarily bounded sets

Pub Date : 2022-04-03 DOI:10.1002/malq.202100020

Emil Jeřábek

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引用次数: 1

Abstract

We show that for any $k \in ω$ , the structure $⟨ H_{k}, \in ⟩$ of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure $V_{ω} = ⋃_{k} H_{k}$ of hereditarily finite sets, which is well known to be bi-interpretable with the standard model of arithmetic $⟨ N, +, \cdot ⟩$ .

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遗传有界集理论

我们证明对于任何k∈ω $k\in \omega$，遗传上大小最多为k的集合的结构⟨H k，∈⟩$\langle H_k,{\in }\rangle$是可决定的。我们提供了它的理论的一个透明的完全公理化，一个量词消除结果，以及它的计算复杂性的严格界限。这与遗传有限集的结构V ω = k H k $V_\omega =\bigcup _kH_k$形成鲜明对比，这是众所周知的，用算术⟨N， +，·⟩$\langle \mathbb {N},+,\cdot \rangle$的标准模型是双可解释的。

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