If something is provable in principle, then (with a certain, admittedly contrived and inefficient search algorithm) the proof can be found in finite time with probability 1. No?
Finite amount of time, yes. Feasible amount of time, no, unless P = NP. When I said that you were considering agents with unlimited resources, this is what I meant—agents for whom “in principle” is not different from “in practice.” There are no such agents under the sun.
But you don’t have to have unlimited resources, you just have to have X large but finite amount of resources, and you don’t know how big X is.
Of course, in order to prove that your resources are sufficient to find the proof, without simply going ahead and trying to find the proof, you would need those resources to be unlimited—because you don’t know how big X is. But you still know it’s finite. “Feasibly computable” is not the same thing as “computable”. “In principle” is, in principle, well defined. “In practice” is not well-defined, because as soon as you have X resources, it becomes possible “in practice” for you to find the proof.
I say again that I do not need to postulate infinities in order to postulate an agent which can find a given proof. For any provable theorem, a sufficiently (finitely) powerful agent can find it (by the above diagonal algorithm); equivalently, an agent of fixed power can find it given sufficient (finite) time. So, while such might be “unfeasible” (whatever that might mean), I can still use it as a step in a justification for the existence of infinities.
Finite amount of time, yes. Feasible amount of time, no, unless P = NP. When I said that you were considering agents with unlimited resources, this is what I meant—agents for whom “in principle” is not different from “in practice.” There are no such agents under the sun.
But you don’t have to have unlimited resources, you just have to have X large but finite amount of resources, and you don’t know how big X is.
Of course, in order to prove that your resources are sufficient to find the proof, without simply going ahead and trying to find the proof, you would need those resources to be unlimited—because you don’t know how big X is. But you still know it’s finite. “Feasibly computable” is not the same thing as “computable”. “In principle” is, in principle, well defined. “In practice” is not well-defined, because as soon as you have X resources, it becomes possible “in practice” for you to find the proof.
I say again that I do not need to postulate infinities in order to postulate an agent which can find a given proof. For any provable theorem, a sufficiently (finitely) powerful agent can find it (by the above diagonal algorithm); equivalently, an agent of fixed power can find it given sufficient (finite) time. So, while such might be “unfeasible” (whatever that might mean), I can still use it as a step in a justification for the existence of infinities.