Incompleteness and undecidability
In Episode 1, we introduced the very idea of a negation-incomplete formalized theory T . We noted that if we aim to construct a theory of basic arithmetic, we’ll ideally like the theory to be able to prove all the truths expressible in the language of basic arithmetic, and hence to be negation complete. But Gödel’s First Incompleteness Theorem says, very roughly, that a nice theory T containing enough arithmetic will always be negation incomplete. Now, the Theorem comes in two flavours, depending on whether we cash out the idea of being ‘nice enough’ in terms of (i) the semantic idea of T ’s being a sound theory, or (ii) the idea of odel’s own T ’s being a consistent theory which proves enough arithmetic. And we noted that G¨.
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