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Every proof I've read about this fact considers two cases: $A$ - finite and $A$ - infinite but this is undecidable problem. So, is there constructive proof?

Every proof I've read about this fact considers two cases: $A$ - finite and $A$ - infinite but this is undecidable. So, is there constructive proof?

Every proof I've read about this fact considers two cases: $A$ - finite and $A$ - infinite but this is undecidable problem. So, is there constructive proof?

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Is there constructive proof of the fact that every recursive set $A \ne \varnothing$ is recursively enumerable in non-decreasing order?

Every proof I've read about this fact considers two cases: $A$ - finite and $A$ - infinite but this is undecidable. So, is there constructive proof?