Adam Scherlis comments on The Goldbach conjecture is probably correct; so was Fermat’s last theorem

• If it can’t be proven false, then it definitely isn’t false!

Equivalently: If it’s false, then it can be proven false.

Why do I say that? Well, if it’s false, then there exists a power of two that is the reverse of a power of five. But that has a very short proof: just write down the smallest example.

As to the case where it can’t be proven either way: I would say that it has to be true in that case, but this might be one of those things that sufficiently diehard constructivists would agree with you on.

• “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.