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Theories of arithmetic in first-order logic, such as Peano arithmetic, second-order arithmetic, Heyting arithmetic, and their subsystems and extensions. Models of Peano arithmetic.
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Presburger Arithmetic
In your question you say Presburger Arithmetic "proves its own consistency". Really? It's provably consistent, as the Wikipedia article notes, but isn't the proof done in a metalanguage? Unfortunat …