ESTE BLOG HA SIDO HACKEADO

In this short article we will provide an example where one's intuition can be faulty, and that strict formalization is always needed when one does mathematics in order to avoid possible hidden intuitive inconsistencies. A striking example is the erroneous but intuitively obvious thesis: "if a formal system F proves that there exists proof of formula A in F , then F has proved A " or slightly differently version is this one: "it must be always provable in all formal systems that from formula saying that there exists proof of A in F , indeed follows A in F ". Both formulations of this naive intuition concerning basics of math logic severely fail. We will show that a consistent F cannot prove for arbitrary formula n that from formula that asserts ``exists proof of n '' follows n . This is a major ingredient of a theorem proved by Löb in 1955.Löb's theorem: If FxP(xn)n , where x is the Gödel number of the proof of the formula with Gödel number n , and n is the numeral of the Gödel number of the formula n , then Fn .The proof presented here follows Karlis Podnieks (2006):Suppose FxP(xn)n . Let F be the new proof system that is equal to F plus the new axiom n . By modus tollens FnxP(xn) . Since F has as axiom n by modus ponens FxP(xn) . In this case F proves that n is not provable in F (and also that F is consistent). This however leads to proof that F is also consistent because it contains only F and n , and we already know both that F is consistent and that F does not prove n . Thus F proves its own consistency. According to Gödel's second theorem howe

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ver F cannot prove its own consistency unless being inconsistent. Therefore F+n must be always inconsistent theory, and we conclude that Fn .Now see that the intuitive notion that xP(xn)n is not provable within arbitrary formal system. If the formal system is inconsistent obviously it proves every formula so the above intuitive notion will be provable. If however F is consistent it cannot prove xP(xn)n for arbitrary formula, because if it could, for formally refutable formula k[0=1] , F could prove by modus tollens kxP(xk) , and since k is formally provable formula, then the formal system could infer its own consistency, or said in ordinary language: from k , F could have proved "there exists at least one unprovable formula in F ", which is impossible according to Gödel's second theorem.Corollary: If F is consistent formal system then F cannot prove for arbitrary formula n that from formula that asserts ``exists proof of n '' follows n .The proof of the corollary is straigthforward either directly from Löb's theorem, or by independent reasoning using the fact that consistent F cannot prove xP(xn)k for refutable k .The relationship between the direct text of Löb's theorem in the proof provided by Podnieks, and the corrolary is this one: if the system F is consistent, it might be able to prove xP(xn)n for provable formulas n , because the provability of n taken together with the consistency of F ensure unprovability of n and hence modus tollens reversed formula of xP(xn)n cannot be used for F to prove its own consistency. Also taking into account the second Gödel theorem it is easy to be seen that F should not be able to prove xP(xn)k for disprovable (refutable) formulas k , a result that alone is sufficient to prove the corollary. Still the corollary is a weaker result than Löb's theorem, because the corrolary does not give us clue whether xP(xn)p is provable for undecidable formulas p . This was indeed the "open problem" proposed by Leon Henkin. The Löb's theorem answers this question, and shows that it is impossible for consistent F to prove xP(xn)p for undecidable formulas p . Therefore summarized, Löb's theorem says that for refutable or undecidable formulas , the intuition "if ``exists proof of '' then " is erroneous.

: PICO.