The halting problem also lets you define a noncomputable number: https://math.stackexchange.com/questions/462790/are-there-any-examples-of-non-computable-real-numbers/462835#462835 … (I like this more than the standard examples like Chaitin's constant, because here we can write down the first 14 million digits explicitly so it feels more like a number than Chaitin's...
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... constant which feels like a linguistic trick when you first see it.)
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