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D and W are expressed as first order sentences.
We may consider second order logic in which we can define a predicate N x) for the set of natural numbers and furthermore can prove of a new predicate symbol Tr(x) that it satisfies the inductive clauses of the truth definition of first order formulas of arithmetic, relativized to N. Then the stronger system can prove that provable first order sentences of arithmetic satisfy the predicate Tr.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com