Computability Properties
As discussed above, Th is not arithmetically definable, by Tarski's theorem. A corollary of Post's theorem establishes that the Turing degree of Th is 0(ω), and so Th is not decidable nor recursively enumerable.
Th is closely related to the theory Th of the recursively enumerable Turing degrees, in the signature of partial orders (Shore 1999:184). In particular, there are computable functions S and T such that:
- For each sentence φ in the signature of first order arithmetic, φ is in Th if and only if S(φ) is in Th
- For each sentence ψ in the signature of partial orders, ψ is in Th if and only if T(ψ) is in Th.
Read more about this topic: True Arithmetic
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