AI 中文总结
本文在PA的无参数可解释性下分类弱集合理论，确定其一致性强度，将其分为三类并说明后续研究方向。
AI 中文摘要
我们在普通无参数可解释性下，根据与它们相互可解释的一阶算术理论，对一大类弱一阶集合理论进行分类，这也确定了它们的一致性强度。对应同一算术理论的集合理论常通过演绎扩展相关，因此只需在算术理论中解释一个更强的集合理论，并在较弱的集合理论中恢复该算术理论，所有中间情形即可推导得出。所研究的集合理论大致分为三类：既无幂集也无穷公理的理论、有幂集但无穷公理的理论、有无穷公理但无幂集的理论。最后简要说明有待研究的更高层级内容。
英文摘要
We classify a broad family of weak first-order set theories, under ordinary parameter-free interpretability, by the first-order arithmetical theories with which they are mutually interpretable. This also determines their consistency strength. Set theories that correspond to the same arithmetical theory are often related by deductive extension. It is therefore enough to interpret a stronger set theory in the arithmetical theory and to recover the arithmetical theory in a weaker set theory; all intermediate cases then follow. The set theories under consideration fall roughly into three classes: theories with neither Power Set nor Infinity, theories with Power Set but without Infinity, and theories with Infinity but without Power Set. We conclude with a brief account of the higher levels that remain to be investigated.
