Is there any known consistent but incomplete formal axiomatic system apart of and simpler than one "capable of doing arithmetic"? Is it even possible?
Even if this capability of arithmetic were a necesary condition, then I think it can't be proven that it's the minimal formal axiomatic system for doing that, so there must be many examples of smaller and smaller system that allow it.
I am not asking for "undecidable mathematical statements" but minimal incomplete systems.
Thanks for any guide