Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
So I know that a conjecture/statement or negation of it plus ZFC can both turn out to be consistent, which means that a statement is not provable. But I would like to go opposite way - and let's say ...
In one of my lectures, the lecturer put a bunch of examples of sets on the board, stuff like the set of all humans, set of all well typed programs in some programming language, the set of all true ...
Is there a mechanical procedure within PA to reduce any ϕ to its simplest (in terms of the arithmetical hierarchy) logically equivalent form?
The question is motivated because we know that the Turing computable sets of natural numbers are exactly the sets at level $\Delta_1^0$ of the arithmetical hierarchy. We also know that the finite ...