The short answer that I know is positive. Let’s talk about PA. It has a model, namely the set of natural numbers, hence it’s consistent. Likewise ZFC is consistent in a larger model. Well, that’s a proof, which I don’t fully understand, but does that mean you can’t prove 1 = 0 in PA forever? There are counter-evidences as well. First a US mathematician tried to prove the inconsistency of arithmetic, I think it’s Ed Nelson. But Terence Tao later found a bug. But if the above proof works so well, why did he try at all? That’s one. Second, the short Buddhist central text Heart Sutra claims, a Bodhisattva after deep meditation, realized that forms are empty and emptiness is forms. I tend to interpret that as 1 = 0. In other words, early Buddhist devotees have proved the inconsistency of arithmetic. There are other places in the Buddhist literature mentioning a sphere that gives all treasures, that is, the contradiction gives all theorems. And the Buddha’s wisdom is supreme, the highest. If there is nothing higher, it must be the contradiction. Anyway, everything to me points to the possibility that Buddhists achieved this mathematical feat in the past, before establishing the religion. So I’m really interested whether the inconsistency of arithmetic is a valid possibility and if so, whether we shall try to attain it as a research goal. Yes it’s super difficult as it implies all, but that also makes it very interesting and relevant. Another evidence is the law of large numbers or what I call the crowd effect, or quantitative change becomes qualitative after a threshold. Anyway it’s interesting and something for us to investigate further.