We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.


We can also agree that we're sure to find more non-provable statements in ZFC. What I'm curious about is the following:


What is the defining characteristic of a non-provable statement in ZFC? Are they all "strong" in some sense? Is it a necessary condition that they are strong, then? What future theorems might turn out to be non-provable? Is it, through the characteristics of the theorems known to be non-provable possible to make a fairly accurate guess if a theorem will turn out to be non-provable in ZFC?



When is a statement provable?
Not all statements that are not provable in ZFC are "strong", if by strong you mean that
ZFC + the statement in question is stronger than ZFC in the sense that it implies the consistency of ZFC.  

The typical example is the Continuum Hypothesis (CH). ZFC + CH is consistent iff ZFC is,
and in this sense, CH is not strong since ZFC + CH itself does not imply the consistency
of ZFC. 


However, if one wants to prove that the existence of a measurable cardinal is not provable in ZFC, the argument is that ZFC + there is a measurable cardinal implies the consistency of 
ZFC and hence, by the second incompleteness thm, the existence of a measurable cardinal cannot
be proved in ZFC. 


Note that the situation in set theory is different from number theory.
In order to show that something is not provable in Peano Arithmetic (something which is consistent with PA), one usually uses the Incompleteness Theorems, i.e., 
one shows that the statement implies the consistency of PA.
So in some sense, number theoretic statements known to be independent over PA 
are strong (over PA), statements known to be independent over ZFC are not necessarily strong over ZFC.


