Not the first time Goedel's Incompleteness theorem has been related to consciousness. It's a running theme in Douglas Hofstadter's work, especially in Goedel, Escher, Bach.

I think Hofstadter's answer to this question would be: To what extent is AI engaging in and using self-reference? That's the degree of consciousness it has.

https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop