Incompleteness and undecidability
| Abstract | 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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Similar books and articlesG. Longo (2011). Reflections on Concrete Incompleteness. Philosophia Mathematica 19 (3):255-280.
Panu Raatikainen (1998). On Interpreting Chaitin's Incompleteness Theorem. Journal of Philosophical Logic 27 (6):569-586.
Peter Smith (2007). An Introduction to Gödel's Theorems. Cambridge University Press.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Laureano Luna & Alex Blum (2008). Arithmetic and Logic Incompleteness: The Link. The Reasoner 2 (3):6.
Raymond M. Smullyan (1993). Recursion Theory for Metamathematics. Oxford University Press.
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