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Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this volume constitutes a selection of papers presented at the Internatonal Conference on Infinity in Logic and Computation, ILC 2007, held in Cape Town, South Africa, in November 2007. The 7 revised papers presented together with 2 invited talks were carefully selected from 27 initial submissions during two rounds of reviewing and improvement. The papers address all aspects of infinity in automata theory, logic, computability and verification and focus on topics such as automata on infinite objects; combinatorics, cryptography and complexity; computability and complexity on the real numbers; infinite games and their connections to logic; logic, computability, and complexity in finitely presentable infinite structures; randomness and computability; transfinite computation; and verification of infinite state systems.