Can you prove the unprovable?

I'm only starting with logic, so I can't say for certain. There is however an interesting line in the wikipedia page:

http://en.wikipedia.org/wiki/Independence_%28mathe...

"The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, [...] they cannot be proved in ZFC [...] :

The existence of strongly inaccessible cardinals

The existence of large cardinals"

which seems to be an affirmative answer to your question, at least in the two specific cases given there, unless it turns out that the statements are actually false.

As scythian says, you may need to be more careful about whether the statement is true; for number theoretic statements and proofs we might turn to a model of the axioms, but with set theory this might seem counter to our goal.

Wow what a question?

I am not sure I can answer your exact question without some research but here is some related info that might be of interest.

When I went through Godel's Incompleteness Theorem I got the feeling it was a retelling of the problem of ultimate causation or ultimate foundation.

As a computer programmer I know that all software code is based on existing procedures that ultimatly have hardwired actions behind them.

Something like Godel's Incompleteness is seen in Modal Logic's Possible Worlds semantics mathematics. In Possible Worlds mathematics world views are modelled by a language that represents definition of sets of facts, similar to a computer program or a dictionary that describes our real world. When we add a rule (e.g. a new defined word) we find we can describe or make clear some hidden aspect that could not previously be described previously. We also however find that new rules also close off possibilities. E.g. If we define an immovable object we can not have an irristable force in the same universe.

Putting aside the foundation problem where all programs need a core set of instrutions, Godellian limitations do not stop self referencial and fuzzy computer programs from working.

I think it was Richard Dreyfus who suggested human minds or souls were superior to machines because computers could only generate incomplete or inconsistent models of the world but my experience is humans arfe also incomplete and inconsistent in their thinking.

I suspect Information Physics and modern quantum information theories suggest that our universe may not only be incomplete and/or inconsistent but may be necessarily so.

If Godellian Incompleteness applies to physics, physical worlds as we seem to experience may be impossibilities, and nothing more than useful lies (or pargmatic truths).

Look for a section something like "Impossible World" on my website megaphysics.info for more.

Godel also looked into selfreferencial time travel. Google "Godelian Time Travel" for more details.

Email me for more info, or post me some new ideas to post on mhy Godel section on Megaphysics.Info

If it turns out that logic is illogical or our universe is impossible then our world is like a set of assumtions that contradict, where 1+2= 3 can be proven bith both true and false.

I hope this helps a little,

Graham P

At least one of us might be confused about this matter. Godel's first incompleteness theorem states that there exists some statements that are true but unprovable, which is the specific case you're addressing. However, if one then makes the following statement, to be proved true or false, "This Statement A is one example of a statement that is true but unprovable", then this itself can also be an undecidable, i.e., a proof either confirming or denying it cannot be expected to exist. Godel's Incompleteness works makes no effort to estimate the LIKELIHOOD of the existence of a proof. For example, if one makes a statement in Euclidean geometry that seems reasonably well-defined, a proof of some sort (for or against) is very likely, but Godel does not address such estimates of success.

Your suggestion that this be "extended" to show that there are "true statements which inherent unprovability is also unprovable" has a pretty serious error of assumption---namely that to even try to prove that such true statements are unprovable PRESUMES that they are true to begin with! How does anyone know? It's more accurate to say that mathematicians have proven that certain statements are undecidable, i.e., they still cannot deem them to be true or false---there is no way, it is undecidable. But the value of such statements that have shown to be undecidable is that they can possibly show the way to alternate theories in mathematics that seems to be reasonably self-consistent. A classic example of that is Euclid's Parallel Postulate. Because mathematicians had tried to prove it without success, such failure implied that it might be an undecidable, and so some mathematicians decided to assume otherwise, and thus was born non-Euclidean geometries.

Perhaps some other expert can come in and clarify things for us here?

No, because if you could, it would not have been unprovable.