Show that there is no recursive set such that and , the complement of . (This result can be stated: The theorems of cannot be recursively separated from the refutable sentences.) Suggestion: Make a sentence saying "My Gödel number is in ." Look to see where is.
The definition of is given in the previous slides (cs.nyu.edu/courses/fall09/G22.2390-001/lec/lec11_h4.ps) and the Enderton textbook (page 203), while denotes a set of consequences of .
To start this problem, if says "My Gödel number is in .", then
how do I see where is?