集合的编码概念
The Coding Conception of Set
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中文总结 AI 辅助
该研究提出序数和集合的编码概念，以康托尔原则为基础生成有界序数集及理论$SC^{reg}$。将其扩展到全集理论产生两个不一致的一阶集合理论，证明幂集二分法不可避免，得出$ZFC^ - +$“每个基数都有后继”是无争议片段，幂集公理地位待决。
中文摘要 AI 辅助
我们提出了序数和集合的编码概念，它以康托尔的三个生成原则为唯一基础。有界序数集通过一个双射编码函数与序数本身同步生成，该函数在每个阶段仅选择后继、极限和限制原则实际所需的有限多个有界集。这种选择性编码产生了一阶理论$SC^{reg}$，我们证明它是序数的元数学上正确的理论：它与$ZFGC^+$是双向可解释的，但对集合的一般概念没有断言。通过具有算术和类理解的一元二阶序数理论将该概念扩展到全集理论，根据不同的极大性直觉产生了两个相互不一致的一阶集合理论：一个A型全域$MC_A$，其中每个序数的幂集是一个集合且全域满足$ZFC$；以及一个B型全域$MC_B^+$，其中集合严格多于序数且存在一个“大基数”，超过该基数幂集仍不可编码。我们证明即使在势主义下，这种幂集二分法也是不可避免的，并得出$ZFC^ - +$“每个基数都有后继”是任何真正集合理论唯一在哲学上无争议的共同片段；完整幂集公理的地位仍然是唯一开放的哲学选择点。
英文摘要
We propose the Coding Conception of ordinals and sets, which takes Cantor's three generating principles as its sole foundation. Bounded sets of ordinals are generated synchronously with the ordinals themselves through a bijective encoding function that, at each stage, selects only the finitely many bounded sets actually required by the successor, limit, and restriction principles. This selective coding yields the first-order theory $SC^{reg}$, which we establish is the metamathematically correct theory of the ordinals: it is bi-interpretable with $ZFGC^+$, yet makes no claim about the general concept of set. Extending the conception to full set theory via a monadic second-order ordinal theory with arithmetic and class comprehension produces two mutually inconsistent first-order set theories according to distinct maximality intuitions: a Type-A universe $MC_A$, in which the power set of every ordinal is a set and the universe satisfies $ZFC$; and a Type-B universe $MC_B^+$, in which sets are strictly more than ordinals and a ``largeness cardinal'' exists, beyond which power sets remain unencodable. We prove that this Power Set Dichotomy is unavoidable, even under potentialism, and conclude that $ZFC^-+$``every cardinal has a successor'' is the only philosophically uncontroversial common fragment of any true set theory; the status of the full power-set axiom remains the sole open philosophical choice point.
