>Since this theorem is true for the standard model of the natural numbers but not provable
I've always found the provable vs true comparison confusing.
How can we say the statement is true under the standard model if we cannot prove it? I understand that it could be true, but how do we know it?
If it's proven with second order arithmetic, then this implies it is true under the standard model too?
Or are there statements true independently of the axiomatic system you use to prove them?
(Apologies if this is too off-topic)
I've always found the provable vs true comparison confusing. How can we say the statement is true under the standard model if we cannot prove it? I understand that it could be true, but how do we know it? If it's proven with second order arithmetic, then this implies it is true under the standard model too?
Or are there statements true independently of the axiomatic system you use to prove them? (Apologies if this is too off-topic)