True Theory of Second-order Arithmetic
The true theory of second-order arithmetic consists of all the sentences in the language of second-order arithmetic that are satisfied by the standard model of second-order arithmetic, whose first-order part is the structure and whose second-order part consists of every subset of .
The true theory of first-order arithmetic, Th, is a subset of the true theory of second order arithmetic, and Th is definable in second-order arithmetic. However, the generalization of Post's theorem to the analytical hierarchy shows that the true theory of second-order arithmetic is not definable by any single formula in second-order arithmetic.
Simpson (1977) has shown that the true theory of second-order arithmetic is computably interpretable with the theory of the partial order of all Turing degrees, in the signature of partial orders, and vice versa.
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