Doesn’t such research presume a formal definition?
No. Not an unambiguous one, at any rate.
What possible ambiguity could exist in my quasi-definition?
The usual definition of natural numbers is that every natural number has a successor, only 0 has no predecessor, any two numbers with the same successor are equal, and induction works. This is not sufficient to prevent you from having an infinite chain of “natural numbers” extending in both directions.
If I remember correctly, the idea is that you can construct a sentence that can neither be proven nor disproven. There is another theorem that I’m less familiar with that says that there’s a model of the natural numbers where it’s true, and another model where it’s false. You can add this sentence as an axiom, but you can still construct another such sentence. Any computable, consistent axiom schema will have such a sentence. Even if it’s as complicated as “simulate 3^^^3 mathematicians for 3^^^3 years and any axioms and computable axiom schemas that they think up”.
I think it’s Gödel’s sentence. It’s basically that you find a way to encode a sentence into a number, then you find some crazy mathematical sentence that basically claims a given number corresponds to an unprovable sentence. You make another sentence that results in the truth value of the given sentence being plugged into itself. By plugging the number for the first sentence into the second, you have a sentence that states that it’s false.
You can say that Gödel’s sentence is false, and the one that’s made with that, etc. are all false, but that axiom schema introduces a new sentence. You can do that again, but then there’s another level. You can do it for an infinite number of levels, but there’s another level after that. You can only do so much recursion.
Doesn’t such research presume a formal definition? What possible ambiguity could exist in my quasi-definition?
You appear to be contradicing yourself regarding human intuition.
No. Not an unambiguous one, at any rate.
The usual definition of natural numbers is that every natural number has a successor, only 0 has no predecessor, any two numbers with the same successor are equal, and induction works. This is not sufficient to prevent you from having an infinite chain of “natural numbers” extending in both directions.
If I remember correctly, the idea is that you can construct a sentence that can neither be proven nor disproven. There is another theorem that I’m less familiar with that says that there’s a model of the natural numbers where it’s true, and another model where it’s false. You can add this sentence as an axiom, but you can still construct another such sentence. Any computable, consistent axiom schema will have such a sentence. Even if it’s as complicated as “simulate 3^^^3 mathematicians for 3^^^3 years and any axioms and computable axiom schemas that they think up”.
I’d be curious to see what said sentence is, and what said proof is.
I think it’s Gödel’s sentence. It’s basically that you find a way to encode a sentence into a number, then you find some crazy mathematical sentence that basically claims a given number corresponds to an unprovable sentence. You make another sentence that results in the truth value of the given sentence being plugged into itself. By plugging the number for the first sentence into the second, you have a sentence that states that it’s false.
You can say that Gödel’s sentence is false, and the one that’s made with that, etc. are all false, but that axiom schema introduces a new sentence. You can do that again, but then there’s another level. You can do it for an infinite number of levels, but there’s another level after that. You can only do so much recursion.