Gödel established fundamental limits on a very specific notion of “knowing”, a proof, that is, a sequence of statements that together justify a theorem to be true with absolute certainty.
If one relaxes the definition of knowing by removing the requirement of absolute certainty within a finite time, then one is not so restricted by Gödel’s theorem. Theorems regarding nonfractional numbers such as what Gödel used can be known to be true or false in the limit by checking each number to check whether the theorem holds.
Theorems of the nature “there exists a number x such that” can be initially set false. If such a theorem is true then one will know eventually by checking each case. If it is not true, then one is correct from the start. Theorems of the nature “for all numbers x P(x) holds” can be initially set true. If such a theorem is false then one will know eventually by checking the cases. If such a theorem is true then one is correct from the start.
The limitation here is absolute certainty within a finite time. One can be guaranteed to be correct eventually, but not know at which point in time correctness will occur.
Gödel. And no the halting problem is separate from Gödels arguments.
Gödel established fundamental limits on a very specific notion of “knowing”, a proof, that is, a sequence of statements that together justify a theorem to be true with absolute certainty.
If one relaxes the definition of knowing by removing the requirement of absolute certainty within a finite time, then one is not so restricted by Gödel’s theorem. Theorems regarding nonfractional numbers such as what Gödel used can be known to be true or false in the limit by checking each number to check whether the theorem holds.
Theorems of the nature “there exists a number x such that” can be initially set false. If such a theorem is true then one will know eventually by checking each case. If it is not true, then one is correct from the start. Theorems of the nature “for all numbers x P(x) holds” can be initially set true. If such a theorem is false then one will know eventually by checking the cases. If such a theorem is true then one is correct from the start.
The limitation here is absolute certainty within a finite time. One can be guaranteed to be correct eventually, but not know at which point in time correctness will occur.