What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-verifying theory?
What about the second theorem? Given that this is a stronger result I would expect the axiomatic systems to be different.
EDIT (clarification): What's the minimal (plausibly) consistent theory needed to prove the First Incompleteness theorem about some theory T, including a sufficient fragment of Robinson arithmetic Q,