Not all ordinals are computable, right?
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Right. You would need a computable upper bound for the ordinals that can be used.
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Why does the upper bound need to be computable?
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Let X be set of ordinals expressible in some data type for ordinals such that < is computable and a∈X ∧ b<a → b∈X. Then sup(X) is computable
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Proof: the program computing < on that data type is a computation of sup(X).
End of conversation
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