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Abstract
Dynamic topological logic is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces.
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DOI 10.1093/jigpal/jzaa055
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References found in this work BETA

On Axiomatizability Within a System.William Craig - 1953 - Journal of Symbolic Logic 18 (1):30-32.
Dynamic Topological Logic.Philip Kremer & Grigori Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
Dynamic Topological Logic.Philip Kremer & Giorgi Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
A Model Existence Theorem in Infinitary Propositional Modal Logic.Krister Segerberg - 1994 - Journal of Philosophical Logic 23 (4):337 - 367.

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