Let be a recursive set of sentences in a recursively numbered language with and . Assume that every recursive relation is representable in the theory . Further assume that is -consistent; i.e., there is no formula such that and for all , . Construct a sentence indirectly asserting that it is not a theorem of , and show that neither nor .

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(Suggestion: See Section 3.0.)

Remark: This is a version of the incompleteness theorem that is closer to Gödel's original 1931 argument. Note that there is no requirement that the axioms be in . Nor is it required that include ; the fixed point argument can still be applied.

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If says that it is not a theorem of , then it is clear that . How do I also show ?

Another question is why -consistent is considered here?

Thanks.