Reflecting on logical dreams

The independence phenomenon The “forcing method,” introduced by Cohen (1966), has produced a wealth of independence results in set theory. What about PA, that is, arithmetic? Now Shelah (2003a) described the following Dream 2.1: Find a “forcing method” relative to PA which shows that PA and even ZFC does not decide “reasonable” arithmetical statements, in analogy with the situation in which the known forcing method works for showing that ZFC cannot decide reasonable set theoretic questions; even showing the unprovability of various statements in bounded arithmetic (instead of PA) is formidable. Why is it interesting to prove independence results? I can understand someone disregarding the work on cardinal arithmetic, claiming that it is not interesting, though I think he or she is wrong. If I had to act as a lawyer I could try to write an argument for him or her; I could express myself as “(s)he is wrong but consistent.” But in this case it is hard for me to understand any opposition (well, from mathematicians, let us say pure mathematicians).