The theory of true arithmetic consists of all first-order arithmetic sentences true in the standard natural-number structure. It is complete but not computably enumerable.
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True arithmetic generally refers to the concept or philosophical position that arithmetic statements can be assigned a truth value based on whether they accurately describe the structure and properties of mathematical objects, particularly in the context of number theory and logic. In more formal terms, true arithmetic often pertains to the set of arithmetic truths that can be validated within a certain logical framework, such as first-order logic or axiomatic systems like Peano arithmetic.