Wittgenstein on Godel's Proof
Did Wittgenstein misunderstand Gödel's first incompleteness theorem?
|Wittgenstein on Godel's Proof|
aka I. Kabruob
Joined: May 19, 2005
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Posted Jul 30, 2012 - 12:45 PM:
DanLanglois wrote:No, if the extension is recursively axiomatizable and consistent, then it is incomplete.
We can extend the system in a way that allows a formal proof of the proposition that was not provable in the original system. I infer that the system is complete
DanLanglois wrote:It means to add symobls, axioms, or rules of inference.
But then what does 'extend the system' mean?
But it's easier to speak of theories rather than systems. With a certain definition, a theory is a set of sentences closed under entailment. Then (for first order) a theory is the set of theorems of a system. Then a theory T* is an extension of a theory T when T is a subset of T*. And T* is a proper extension of a theory T when T is a proper subset of T*.