The Theorem and its proof is given below:

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But I am wondering why we write "Ignore the input " when describing an Enumerator? could anyone explain this for me please?

  • $\begingroup$ Please credit the source of the quoted text. (It would also be nice if you transcribed the text itself to make the question searchable and accessible to people with visual impairments, but I realise that's a much bigger ask.) $\endgroup$ – David Richerby Jun 4 '18 at 17:29
  • $\begingroup$ OK I will add the source @DavidRicherby $\endgroup$ – Intuition Jun 4 '18 at 18:22

It's just an explicit statement that the behaviour of the machine doesn't depend on its input in any way. This could be inferred from the rest of the machine's description, but explicitly saying it makes it clear that it's not a mistake.

  • $\begingroup$ But the enumerator prints the input so the behavior of the enumerator depend on its input ...... or what is your opinion? $\endgroup$ – Intuition Jun 4 '18 at 18:26
  • $\begingroup$ No, the enumerator prints the strings from the language, one after another. (With some conditions to ensure "proper" behaviour for infinite languages.) $\endgroup$ – David Richerby Jun 4 '18 at 18:29

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