Lower order axioms

[JOSEPH SHIPMAN]

What natural axiom systems *for arithmetic* extend the Peano axioms?

Obviously I can define an axiom scheme “if Phi is an arithmetical consequence of ZF, then Phi” and I’ll get any provable arithmetical statement, but what I really want to know is if there are any standard systems stronger than PA in Second-Order Arithmetic or a higher system, such that the purely arithmetical consequences follow from a nice set of first-order arithmetical axioms. 

Analogously, is there a natural axiom system in the language of Second Order Arithmetic with the strength of a higher order system?

— JS

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