I am sorry from not being able to write in Hebrew from here.

I am now at a Set-Theory conference in Barcelona and had the luck to meet the legendary set-theorist Mathias (he has some kind of real numbers called on his name).

I asked him about this Goldbach Conjecture and he told that since it is an assertion that is logically as complex as the assertion that ZFC is consistent, a priory it *could* be the case that it is independent of ZFC.

So all I said before is nonsence. And this is a good time to do "Tshuva" on this (which I just do...).

Now, one can give a meta-mathematical argument that if GC is consistent, then it is true in the *standard* model of set theory. Something like this was suggested by Vishne, et. al., in other correspondenses here.

But this does not prove that it is true. The theory of the natural numbers has models with "infinite" elements which code things that locally look like proofs but are not really proofs because they are infinite.

I am sorry for being vague – I am not an expert on these and tried to write something close to what I heard from Professor Mathias.

He told me that there exist arithmetic theorems which, unlike the Paris-Harrington case where actually the Ramsey Theoretic assertion is provable in ZFC, are undecideable in ZFC. He referred me to Harvey Friedman´s homepage. Google fount it at:

http://www.math.ohio-state.edu/foundations/

I hope that everything you want can be found there.

Shana Tova,

Dr. T.

10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain "why" they occur, or fails to provide a "mechanism".