> A theorem that is undecidable (neither provable nor refutable) from a set of axioms cannot be 'true' in the logical sense (because there are models of that set of axioms in which the theorem is true, and other models in which the theorem is false)
Yes, fair point. A more careful way to make the point I was trying to make is that you might care about the models in which the undecidable statement is true.
E.g. we certainly care about geometries in which a pair of lines that cross another line at the same angle are truly parallel, such that they do not meet.
We care about this even though it's not provable from four other postulates of geometry.
Yes, fair point. A more careful way to make the point I was trying to make is that you might care about the models in which the undecidable statement is true.