Torkel Franzén has written an informative and amusing account of this phenomenon.Īnybody who does anything involving Gödel’s theorems quickly gets contacted by cranks who inform them that their work is wrong. Beyond such technical points as these, most remarks on the consequences of the incompleteness theorems (even by some serious academics) are complete bullshit. Gödel’s theorem was unwelcome when it was announced: it destroyed at a stroke Hilbert’s programme for putting mathematics on a sound footing by proving the consistency of strong formal systems (such as Zermelo–Fraenkel set theory) in a weaker system (such as Peano arithmetic). The second incompleteness theorem states that, in particular, no statement implying the consistency of $F$ is provable in $F$. Moreover, $G_F$ will be true in the standard model for $F$. one that is neither provable nor disprovable in $F$. Gödel’s first incompleteness theorem states that for any “reasonable” formal system $F$ there exists some undecidable statement $G_F$, i.e. Don’t they immediately imply that any project to formalise mathematics is doomed to fail? An overview of Gödel incompleteness Gödel’s incompleteness theorems are often regarded as placing strict limits on the power of logic. General logic incompleteness David Hilbert Kurt Gödel Do Gödel's incompleteness theorems matter? Machine Logic At the junction of computation, logic and mathematics Do Gödel's incompleteness theorems matter?
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