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.
In any case, do you have some alternative theory of arithmetic in mind? Being able to form questions that it can’t prove an answer to seems like much less of a problem (and a problem that a non omniscient AI needs to solve anyway) than not being able to do arithmetic.
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).
We should preserve as much as possible, but no more. Giving up infinity means, you have no trouble with Godel’s incompleteness anymore.
Gödel’s incompleteness theorems apply to any system that contains peano arithmetic. Infinity doesn’t come into it.
Peano arithmetic implies infinity. The infinite set of all natural numbers.
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.
In any case, do you have some alternative theory of arithmetic in mind? Being able to form questions that it can’t prove an answer to seems like much less of a problem (and a problem that a non omniscient AI needs to solve anyway) than not being able to do arithmetic.
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).