Primitive recursion, but using ordinals as bounds instead of integers (ie loops have an ordinal variable which must decrease each iteration)
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Proof: the program computing < on that data type is a computation of sup(X).
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Why couldn't we use a set X that violates the second assumption?
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You could, but then the order type of X would still be computable for the same reason, so it doesn't let you do anything different.
End of conversation
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