Would it be possible to use a paraconsistent logic and axioms similar to ZFC to create a formal sytem, that can be proven to be non-trivial (so that there are some statements which can´t be proven in the system), and which can serve as a foundation for mathematics?


Because it´s possible that the current ZFC + first order logic foundation is inconsistent, and if it´s inconsistent then it would be completely trivial and worthless. And according to my understanding of gödel´s second incompleteness theorem it´s impossible to show that ZFC is consistent (at least without using a formal system that´s even more questionable). This means it´s impossible to prove that the current ZFC foundations are non-trivial, and the same holds for any other foundation that uses first order logic and contains basic arithmetic.


Naturally, I think it would be more desirable to have a foundation that is demonstrably non-trivial, than a foundation where we can´t possibly prove non-triviality. That´s why I wonder whether a paraconsistent foundation could be demonstrably non-trivial.


So basically I have two questions:


1. Is it possible to have a non-trivial paraconsistent formal system containing basic arithmetic, whose non-triviality can be proven in "weaker" formal systems?
2. If such a system exist, could it possibly be a reasonable foundation for mathematics?



Is a paraconsistent and provably non-trivial foundation for math possible?
There's basically no reason to think that ZFC is inconsistent. Even if it is inconsistent, there's an attractive fragment called [Peano arithmetic](https://en.wikipedia.org/wiki/Peano_axioms), which seems self-evidently consistent to me. Gödel's incompleteness still holds here, and it holds in even weaker fragments of arithmetic. It's because as soon as you have a fragment of arithmetic large enough to do anything interesting, you can encode proofs as numbers in that arithmetic, and then you can derive Gödel's results. This seems intrinsic to the nature of arithmetic to me -- proofs are strings of symbols that we can store in a computer, so of course we can represent it as a big number.


That said, people have attempted things in the direction you suggest. One approach is what's called [relevance logic](http://plato.stanford.edu/entries/logic-relevance/). There's a version of arithmetic for relevance logic that is provably non-trivial. My impression is that it only captures a small fragment of arithmetic, though. My link provides additional details, including references for relevance arithmetic.


