>The truth of a proposition P, within F, means "P is an axiom of F, or can be proved in F".
No, truth and proof are NOT the same thing. The system F needn't have any notion of truth. For example, Peano arithmetic has no notion of truth. Peano arithmetic has notions of zero, successor, addition, and multiplication--none of truth.
Almost all competent mathematicians would agree that there exist theorems in the language of PA which are neither provable nor disprovable in PA but which are nevertheless true. Here, "true" means "true in the standard model of PA", not "true according to PA"---the latter isn't even meaningful because there is no such thing as "truth according to PA".
1. The original example would be Goedel's own example, namely, the statement (formulated in the language of PA) that PA is consistent, i.e., that PA does not prove 1=0.
No, truth and proof are NOT the same thing. The system F needn't have any notion of truth. For example, Peano arithmetic has no notion of truth. Peano arithmetic has notions of zero, successor, addition, and multiplication--none of truth.
Almost all competent mathematicians would agree that there exist theorems in the language of PA which are neither provable nor disprovable in PA but which are nevertheless true. Here, "true" means "true in the standard model of PA", not "true according to PA"---the latter isn't even meaningful because there is no such thing as "truth according to PA".