This prompts the question about the truth-value of the statements that are undecided by ZFC. Should one be content with them being undecidable? Does it make sense at all to ask for their truth-value? There are several possible reactions to this. One is the skeptic’s position: the statements that are undecidable in ZFC have no definite answer; and they may even be inherently vague. Another, the common one among mathematicians, is Gödel’s position: the undecidability only shows that the ZFC system is too weak to answer those questions, and therefore one should search for new axioms that once added to ZFC would answer them. The search for new axioms has been known as Gödel’s Program. See Hauser (2006) for a thorough philosophical discussion of the Program, and also the entry on   large cardinals and determinacy  for philosophical considerations on the justification of new axioms for set theory.