Can second order peano arithmetic prove that first order peano arithmetic is sound?
Note that I'm not just talking about its axioms, but also its theorems.
2026-02-22 18:12:02.1771783922
Can second order peano arithmetic prove that first order peano arithmetic is sound?
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Yes. Second-order Peano arithmetic can create the truth predicate for sentences of first-order Peano arithmetic, and verify that every theorem of first-order PA is true under that predicate. In this way, second-order arithmetic proves the consistency of first-order PA, and proving the consistency of a theory is how we normally look at proving a theory to be sound.