Chain Logic and Shelah's Infinitary Logic
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Publication:6323146
DOI10.1007/S11856-021-2207-0arXiv1908.01177MaRDI QIDQ6323146
Jouko Väänänen, Mirna Dzamonja
Publication date: 3 August 2019
Abstract: For a cardinal of the form $kappa=�eth_kappa$, Shelah's logic $L^1_kappa$ has a characterisation as the maximal logic above $�igcup_{lambda<kappa} L_{lambda, omega}$ satisfying Strong Undefinability of Well Order (SUDWO). SUDWO is a strengthening of the Undefinability of Well Order (UDWO). We prove that if $kappa$ is singular of countable cofinality, Karp's chain logic cite{Karpintroduceschain} is above $L^1_kappa$, while it is already known that it satisfies UDWO and Interpolation. Moreover, we show that in these circumstances, the chain logic is -- in a sense -- maximal among logics with chain models to satisfy UDWO. We then show that the chain logic gives a partial solution to Problem 1.4. from Shelah's cite{Sh797}, which asked whether for $kappa$ singular of countable cofinality there was a logic strictly between $ L_{kappa^+, omega}$ and $L_{kappa^+, kappa^+}$ having Interpolation. We show that modulo accepting as the upper bound a model class of $L_{kappa, kappa}$, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not $kappa$-compact, a question that we have asked on various occasions. We contribue to the further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic $L^1_kappa$ in satisfying Interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a Completeness Theorem.
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