Note that we have good reasons for believing that PA is consistent. Gentzen’s theorem is the most obvious one.
Gentzen’s theorem assumes a much stronger claim to prove this weaker one. Yes, it grounds consistency of PA in another mathematical intuition, and in this sense could be said to strengthen the claim a bit, but formally it’s a sham.
(Of course, modern mathematics is much stronger than a century ago, so that alone counts for a good reason to believe that PA is consistent.)
Gentzen’s theorem assumes a much stronger claim to prove this weaker one.
No. The minimal axioms needs to prove Gentzen’s theorem are not stronger than PA. There are claims in PA that cannot be proven in the minimal context for Gentzen’s theorem.
Gentzen’s theorem assumes a much stronger claim to prove this weaker one. Yes, it grounds consistency of PA in another mathematical intuition, and in this sense could be said to strengthen the claim a bit, but formally it’s a sham.
(Of course, modern mathematics is much stronger than a century ago, so that alone counts for a good reason to believe that PA is consistent.)
No. The minimal axioms needs to prove Gentzen’s theorem are not stronger than PA. There are claims in PA that cannot be proven in the minimal context for Gentzen’s theorem.