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The existence of an extension of first order logic satisfying both the Compactness Theorem and the Interpolation Theorem is an open or solved question?

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What properties does an "extension of first order logic" need to satisfy (e.g. relativization, isomorphism-invariance, etc.)? With no restrictions, I think uninteresting examples of such logics can be constructed pretty easily. –  Noah Schweber May 12 '12 at 4:37
Also, can you give any background to this problem? I've heard about it before, and I understand it has an interesting history, but that's the extent of my knowledge about it; and I'd be curious to hear anything you might say about it. –  Noah Schweber May 12 '12 at 4:39

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