Comment by robinhouston

This definitely needs some context for the non-logicians in the house! Gödel's second incompleteness theorem shows that, if PA can prove its own consistency, then PA is inconsistent (and can therefore prove anything, including false things).

The work linked here doesn't show that PA is inconsistent, however: what it does is to define a new, weaker notion of what it means for PA to “prove its own consistency” and to show that PA can do that weaker thing.

Interesting work for sure, but it won't mean anything to you unless you already know a lot of logic.