“If it can’t be proven false, then it definitely isn’t false” Hmm, if you are applying that to mathematical conjectuire, then those statements dont seem compatible with Godel’s theorem to me.
You need to add some assumptions to make it work. For example, I believe the following works:
“In second order arithmetic, we can prove that NP1 implies NF, where NP1 is the statement ‘there exists no first order proof of the conjecture’ and NF is the statement ‘the conjecture isn’t false’.”
I meant this specific conjecture, not all conjectures. More generally it applies to all conjectures of the form “there is no number n such that Q(n)” where Q is straightforward to check for a particular n.
“If it can’t be proven false, then it definitely isn’t false”
Hmm, if you are applying that to mathematical conjectuire, then those statements dont seem compatible with Godel’s theorem to me.
You need to add some assumptions to make it work. For example, I believe the following works:
“In second order arithmetic, we can prove that NP1 implies NF, where NP1 is the statement ‘there exists no first order proof of the conjecture’ and NF is the statement ‘the conjecture isn’t false’.”
I meant this specific conjecture, not all conjectures. More generally it applies to all conjectures of the form “there is no number n such that Q(n)” where Q is straightforward to check for a particular n.