Only if you have some additional theory that says sets exist and have cardinalities and the set of all natural numbers is a valid set. None of which is assumed by the incompleteness theorem.
Peano system demands a successor for every natural. And that 0 (or 1, depends of a variant), is no natural’s successor. So the Peano system is only applicable to infinite sets.
do you have some alternative theory of arithmetic in mind?
Sure. The finite number of possible numbers. As in the computer science already is.
Bounded arithmetic is in some sense trivial; it’s just a big lookup table. I don’t think you’re going to get a useful AI that way; if you’re going to avoid Godel by limiting the complexity of statements the system can consider, it has to be so simple that it can’t form a statement of the form “this statement is not provable” (and not because there’s an explicit rule against such statements, the system has to naturally be that simple).
Peano system demands a successor for every natural. And that 0 (or 1, depends of a variant), is no natural’s successor. So the Peano system is only applicable to infinite sets.
Sure. The finite number of possible numbers. As in the computer science already is.
Bounded arithmetic is in some sense trivial; it’s just a big lookup table. I don’t think you’re going to get a useful AI that way; if you’re going to avoid Godel by limiting the complexity of statements the system can consider, it has to be so simple that it can’t form a statement of the form “this statement is not provable” (and not because there’s an explicit rule against such statements, the system has to naturally be that simple).