> Seriously, we should begin to abandon the idea that whatever is true > or false in the naturals can be structure theoretically proven, > verified. > > If we don't, we'd be in _no_ better position than where Hilbert > was with his All-mighty-formal-system, proving all arithmetic > true formulas. > > We'd be simply change the name "All-mighty-formal-system" > to "All-mighty-language-structure". But it's still an Incompleteness > (of the 2nd kind) that we'd encounter: the Incompleteness of language > structure interpretation of the abstract (non-logical) concept known > as the natural numbers.
But it is known structure theoretically that if we have any 2 structures that satisfy Peano axioms, then they are isomorphic: a statement is true in one if and only if it's true in the other.