Further, is provable if the finite list of formulas encoded by is not a proof of , and is provable if the finite list of formulas encoded by is a proof of . Given any numbers and , either or (but not both) is provable. Any proof of can be encoded by a Gödel number , such that does not hold. If holds for all natural numbers , then there is no proof of . In other words, , a formula about natural numbers, corresponds to "there is no proof of ". We now define the formula , where is a free variable. The formula itself has a Gödel number as does every formula. This formula has a free variable . Suppose we replace it with , the Gödel number of a formula , where is a free variable. Then, corresponds to "there is no proof of ", as we have seen. Consider the formula . This formula concerning the number corresponds to "there is no proof of ". We have here the self-referential feature that is crucial to the proof: A formula of the formal theory that somehow relates to its own provability within that formal theory. Very informally, says: "I am not provable". We will now show that neither the formula , nor its negation , is provable. Suppose is provable. Let be the Gödel number of a proof of . Then, as seen earlier, the formula is provable. Proving both and violates the consistency of the formal theory. We therefore conclude that is not provable. Consider any number . Suppose is provable. Then, must be the Gödel number of a proof of . But we have just proved that is not provable. Since either or must be provable, we conclude that, for all natural numbers is provable. Suppose the negation of , , is provable. Proving both , and , for all natural numbers , violates ω-consistency of the formal theory. Thus if the theory is ω-consistent, is not provable. We have sketched a proof showing that:  For any formal, recursively enumerable (i.e. effectively generated) theory of Peano Arithmetic,  The proof of Gödel's incompleteness theorem just sketched is proof-theoretic (also called syntactic) in that it shows that if certain proofs exist (a proof of or its negation) then they can be manipulated to produce a proof of a contradiction. This makes no appeal to whether is "true", only to whether it is provable.