Presburger arithmetic does NOT prove its own consistency. Its only function symbols are addition and successor, which are not sufficient to represent Godel encodings of propositions.

However, consistent self-verifying axiom systems do exist -- see the work of Dan Willard ("Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles"). The basic idea is to include enough arithmetic to make Godel codings work, but not enough to make the incompleteness theorem go through. In particular, you remove the addition and multiplication function symbols, and replace them with subtraction and division. This is enough to permit representing the theory arithmetically, but the totality of multiplication (which is essential for the proof of the incompleteness theorem) is not provable, which lets you consistently add a reflection principle to the logic.

Presburger arithmetic does NOT prove its own consistency. Its only function symbols are addition and successor, which are not sufficient to represent Godel encodings of propositions.

However, consistent self-verifying axiom systems do exist -- see

• Dan E. Willard, Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles, The Journal of Symbolic Logic. 66 (2) (2001) pp 536–596. doi:10.2307/2695030, JSTOR, (author pdf).

The basic idea is to include enough arithmetic to make Godel codings work, but not enough to make the incompleteness theorem go through. In particular, you remove the addition and multiplication function symbols, and replace them with subtraction and division. This is enough to permit representing the theory arithmetically, but the totality of multiplication (which is essential for the proof of the incompleteness theorem) is not provable, which lets you consistently add a reflection principle to the logic.