/lit/ - Literature

>>22669712
it cannot be proven to be true within peano arithmetic, but it is trivially true when we look at it metalinguistically
the point of godel's theorem is that particular formula systems are incomplete - not (necessarily) that there are universally unprovable things. for example, you can add the Godel sentence to peano arithmetic to get a new formal system within which the original Godel sentence is trivially provable - but this new formal system will have its own Godel sentence thus will be incomplete in its own way.
you wanna know another sentence that isn't provable within a given formal system (containing enough of peano arithmetic)? the sentence stating that the system is consistent. well peano arithmetic is obviously consistent (at least many would contend), so how about we just make the consistency of peano arithmetic an axiom, i.e. by adding the aforementioned consistency sentence to peano arithmetic? sure, let's do that. then we get a new system which cannot prove it's own consistency - but of course, since we believe peano arithmetic is consistent, so must this new one! so let's add the consistency sentence of this system as an axiom. how about we keep repeating this and take the limit of the process, so we obtain a new formal system with a denumerable amount of new axioms stating the consistency of ever more powerful systems. hmm, you're probably wondering what we one can do with this new system. well let me tell what you can do, or more precisely i shall let alan turing tell you! turing proved that in the system constructed in the way just described, one is able to prove all true "universal" statements about the natural numbers - that is, statements of the form "for all natural numbers n, P(n)" for some (i believe unbounded, iirc) proposition P. pic related is mfw first finding this theorem