Computational reasoning about incomputability, infinity, truth etc (Gödel, Tarski,...)

So I would be curious about the theoretical foundations how to make sense of higher-level abstract reasoning like reasoning about infinities, incomputability, truth (which we know cannot be defined due to Tarski) in the field of artificial intelligence. It seems due to Gödel-like constructions you are forced into inconsistent systems of reasoning when operating within a computable system. But those prove everything and "nothing", so as far as I understand it, it kind of upends the whole system of reason …

artificial computational reasoning