The idea of mathematics was always that with simple principles complex things could be explained. This idea has somewhat lost in logic, you need a very complex system to explain something simple.
I am developing a very simple logic, that consists only of context free PI2 sentences. Every theorem is a comparison of two non-deterministic Turing equivalent expressions. I show that is possible to directly reason with such system and that it can be understood by everyone that can program a little bit.
It is not possible to directly reason (without additional logic) about Halting problems. The leap between two Halting problems is too big.
It is also not possible to reason about comparison of two deterministic programs. The leap between two theorems is still too big.
However, with comparison of two non-deterministic programs, things become possible.