There may be limits, depending on your definition of "logic", but there is no part of the incompleteness theorem that exempts the particular type of computer th

There may be limits, depending on your definition of "logic", but there is no part of the incompleteness theorem that exempts the particular type of computer that happened to evolve over the last few millions of years. There is no spark that is fundamentally unreplicable, despite being replicated every time humans reproduce. Art, love, beauty, and morality are parts of our desires that don't correspond neatly to the idea of self-interested maximization of material wealth or whatever, but there is no rule that an AI must have a simple objective function: one could be built with such goals or without them, with useless ones (paperclip maximizer) or weird ones (e.g. complete disregard for the well-being of the self). Hopefully if we ever do build some super AI (I'm not opining on any particular time frame in which this might happen, just on theoretical possibility), it will share a variety of our values, and will therefore want to include such values in any AIs it builds, and so on. But values are fundamentally arbitrary - look at the variety of values in human cultures, and consider how much more alien we should expect a nonhuman intelligence to be.