We generalize the Hart-Shelah example [10] to higher infinitary logics. We build, for each natural number $k\ge 2$ and for each infinite cardinal λ, a sentence ${\psi }_k^{\lambda }$ of the logic $L_{{(2^{\lambda })}^{+},\omega }$ that (modulo mild set theoretical hypotheses around λ and assuming $2^{\lambda }<{\lambda }^{+m}$) is categorical in ${\lambda }^{+},\dots ,{\lambda }^{+k-1}$ but not in ${\mathrm{\beth }}_{k+1}{(\lambda )}^{+}$ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class ${\mathcal{K}}^{⁎}(\lambda ,k)=(Mod({\psi }_k^{\lambda }),{\prec }_{{(2^{\lambda })}^{+},\omega })$ in the finite interval of cardinals $\lambda ,{\lambda }^{+},\dots ,{\lambda }^{+k}$.

中文翻译：

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Hart-Shelah示例，逻辑更强

我们将Hart-Shelah示例[10]推广到更高的无限逻辑。我们为每个自然数建立$ķ\ge 2$对于每个无限基数λ，一个句子${\psi }_{ķ}^{\lambda }$ 逻辑的 ${\mathrm{大号}}_{{（2^{\lambda }）}^{+}，\omega }$那（模温和集合理论假设在λ附近并且假设$2^{\lambda }<{\lambda }^{+米}$）属于以下类别 ${\lambda }^{+}，\dots ，{\lambda }^{+ķ-\mathrm{1个}}$ 但不在 ${\mathrm{\beth }}_{ķ+\mathrm{1个}}{（\lambda ）}^{+}$（或更高）；我们研究了构成该句子的组合词的维数编码，并研究了所得抽象基本类的各种模型理论性质${\mathcal{ķ}}^{⁎}（\lambda ，ķ）=（\mathrm{中号}Ød（{\psi }_{ķ}^{\lambda }），{\prec }_{{（2^{\lambda }）}^{+}，\omega }）$ 在红衣主教的有限区间内 $\lambda ，{\lambda }^{+}，\dots ，{\lambda }^{+ķ}$。