Peano arithmetic is the basis for a lot of math. There are very basic assumptions underlying it, but using just those assumptions you can make the natural numbers and perform math on them.
This was a bold attempt to prove that they are not consistent. That is, you can use the base peano axioms to prove 1 = 0 and anything else you want.
Terry Tao pointed out a mistake the author made, and thus 'saved' mathematics. Peano arithmetic remains consistent.
This isn't quite the conclusion here. Peano arithmetic's consistency cannot be proven within its own limits, so it remains a hope/intuitive belief, but not a fact. Closest we've gotten, to my knowledge, is that there have been consistency proofs within other axiomatic systems:
https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof
I mean, Gödel told us that we can't have a system that proves its own consistency while being consistent, so we shouldn't expect to get closer than "we have proofs in other systems and also the axioms seem so dang simple and obvious and also people have been banging on them and haven't found any inconsistencies..."
But moreover, if Gödel's proof had gone the other way I'm not sure the situation is all that much changed. If I hold in my hand a proof of the consistency of a system of axioms within that system of axioms then... either the system is consistent or, by being inconsistent, could prove anything including its own consistency.
