I am a realist-in-truth-value for recursively-enumerable statements
I am a realist-in-truth-value for recursively-enumerable statements. i.e. I believe that statements for which counterexamples could be found if they existed must be either true or false.
For example, the Goldbach conjecture must be either true or false (since if it were false, you could find a number >4 which is not a sum of two primes). Also, whether any Turing Machine halts is a recursively-enumerable problem.
Basically, it seems to me that any meaningful problem is recursively-enumerable.
A consequence of Gödel's Incompleteness:
If a set of axioms expressing PA is recursively-enumerable, there will be statements which are undecidable.
I believe there should be an algorithmic way of deciding all recursively-enumerable statements, i.e. an algorithmic way of proceeding in making mathematics prove more true things whenever you run into undecidabilities (i.e. whenever you correctly prove, using a meta-method, that a sentence G is true but not provable in the original system). But this seems not to be possible, for if there were any such algorithm, we could construct its halting problem, and *that* problem would be undecidable in the new system.
So I do think that Gödel's theorem is significant for the philosophy of mathematics. And it seems like Chaitin is onto something when he says that mathematics must be semi-empirical.
Again, thanks to Benedikt Löwe for reciting me the theorem above.
For example, the Goldbach conjecture must be either true or false (since if it were false, you could find a number >4 which is not a sum of two primes). Also, whether any Turing Machine halts is a recursively-enumerable problem.
Basically, it seems to me that any meaningful problem is recursively-enumerable.
A consequence of Gödel's Incompleteness:
If a set of axioms expressing PA is recursively-enumerable, there will be statements which are undecidable.
I believe there should be an algorithmic way of deciding all recursively-enumerable statements, i.e. an algorithmic way of proceeding in making mathematics prove more true things whenever you run into undecidabilities (i.e. whenever you correctly prove, using a meta-method, that a sentence G is true but not provable in the original system). But this seems not to be possible, for if there were any such algorithm, we could construct its halting problem, and *that* problem would be undecidable in the new system.
So I do think that Gödel's theorem is significant for the philosophy of mathematics. And it seems like Chaitin is onto something when he says that mathematics must be semi-empirical.
Again, thanks to Benedikt Löwe for reciting me the theorem above.