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Some possible solutions to the paradox come to mind (I have not looked any of this up online, even though I'm sure I'm not the first one...

It has to be a terminating algorithm that computes the number up to a given precision (which can be arbitrarily large). The Cantor's...

Famously, the set of computable numbers is countable. That's pretty much a result of their definition: The decimal expansion of a...