I've been listening to some talks about the continuum hypothesis and I have some questions regarding how we are working on this problem. A particular talk of significance is this one. Here, Woodin talks about the search for new axioms that will "decide" CH. But he speaks about CH as if it has a actual truth value adopting a somewhat platonic perspective of set theory.
Clearly, because CH is independent of ZFC, it makes no sense to talk about its truth value in respect to ZFC. And if we are speaking purely in the universe of ZFC, we can choose whether we want CH to be true or false by just appending another axiom in a arbitrary direction. However, I understand that some mathematicians believe that there are truths about sets outside of our current axiomatization. (Woodin, Gödel, etc.) And I'm don't completely disagree with them. I hold a somewhat plenitudinous platonic view of mathematics; every universe of sets exist per se, but we choose study ZFC because that is what closely reflects our intuitive, abstract concept of what we mean when we say set.
I'm wondering how do we expand our axioms while still talking about the same intuitive sets? I've heard the term forcing thrown around. I'm not quite knowledgable about that process. Does that have to do with expanding the axioms? What is the process of "finding" new axioms like? Listening to Woodin speak, the process seemed very mathematical and not very philosophical. It didn't seem very "soft" either. Does this process require a platonic view of sets to have any meaning at all?
I am vaguely familiar with the concept of large cardinals, and how they're existence is independent from ZFC. We need new axioms to talk about them. What was the process like when adding those axioms? Did we just will them into existence or are there arguments for their existence beyond the order of ZFC?