联合嵌入Hanf数的一个下界
A Lower Bound for the Hanf Number for Joint Embedding
摘要：
在[13]中，作者证明了：如果$μ$是强紧基数，$K$是一个具有$LS(K)<μ$的抽象初等类（AEC），且$K$在$μ$之下满足联合嵌入（合并），则$K$在所有基数$\ge μ$中满足联合嵌入（合并）.提出了强紧上界是否最优的问题。本文证明了在可数词汇表中存在一个可由$\mathcal{L}_{ω_1,ω}$ -语句公理化的AEC $K$，使得如果$μ$是第一可测基数，则（1）$K$在$μ$之下满足联合嵌入cofilm;（2）$K$在$μ$之下不满足联合嵌入cofilm;（3）$K$在$μ$之上满足联合嵌入.此外，该示例可以推广到在大小为$χ$的词汇表中以$\mathcal{L}_{χ^+, ω}$公理化的AEC $K^χ$，使得（1）-（3）保持$μ$是$χ$之上的第一个可测量的。这证明了联合嵌入的Hanf数包含在第一可测和第一强紧之间的区间内。由于这两个基数可以一致地重合，[13]的上限是一致最优的。这也是第一个例子，一个句子的联合嵌入频谱（一贯）既不是一个初始的，也不是一个最终的区间基数。通过定理3.26，一致的是，对于第一可测量$μ$上的任何梅花$C$，JEP在$\lim C$上和$μ$上方的任何地方都精确地成立。
英文摘要：
In [13] the authors show that if $μ$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<μ$, and $K$ satisfies joint embedding (amalgamation) cofinally below $μ$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $\ge μ$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $\mathcal{L}_{ω_1,ω}$-sentence in a countable vocabulary, so that if $μ$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $μ$ ; (2) $K$ fails joint embedding cofinally below $μ$; and (3) $K$ satisfies joint embedding above $μ$. Moreover, the example can be generalized to an AEC $K^χ$ axiomatized in $\mathcal{L}_{χ^+, ω}$, in a vocabulary of size $χ$, such that (1)-(3) hold with $μ$ being the first measurable above $χ$. This proves that the Hanf number for joint embedding is contained in the interval between the first measurable and the first strongly compact. Since these two cardinals can consistently coincide, the upper bound from [13] is consistently optimal. This is also the first example of a sentence whose joint embedding spectrum is (consistently) neither an initial nor an eventual interval of cardinals. By Theorem 3.26, it is consistent that for any club $C$ on the first measurable $μ$, JEP holds exactly on $\lim C$ and everywhere above $μ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
