Archive for September 13th, 2010

Statement of Tarski’s Theorem

September 13, 2010

The theorem showing that the theory determined by the standard model of arithmetic, \mathsf{Th} (\Omega), is undecidable and consequently not decidably axiomatizable (this means that \mathsf{Th} (\Omega) \not= \mathsf{Th} (\Gamma), for some \Gamma consisting of axioms of arithmetic, like those of Peano Arithmetic or one of it’s extensions) can be made stronger by showing that \mathsf{Th}(\Omega)^{\#} := \{p: \chi_{p} \in \mathsf{Th}(\Omega)\},  the set of indices of sentences that are true in \mathsf{Th} (\Omega), is not definable over \Omega, which means that \mathsf{Th}(\Omega) is not effectively enumerable.  This is known as Tarski’s Theorem on the undefinability of truth.

Now set up the (provably effectively computable) function \mathsf{Sb}(m):= \#(\chi_{m}(\dot{m})), which returns the least number m such that \chi_{m} is a formula of \textup{L}_{\Omega}.

In the next update I’ll discuss the proof of Tarski’s Theorem and move into the corollary that Hellman uses to give an example of determination without reduction.

Posted in Tarski's Theorem, Undefinability of Truth | Leave a Comment »

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