I have been thinking about Gödel’s program in set theory for quite a while. Disclaimer: I am no expert in set theory, I am just interested in how Gödel specifically linked the independance of CH with a philosophical background ranging from Kant to Brouwer and from Brouwer to Husserl. Yes, I am sympathetic to the idea that philosophy is not at all useless in mathematics and, more precisely, in the foundations of mathematics (contra, say, what some naturalists claim). Set theory is a mathematical branch as any other, sure, but it is historically clear that many philosophical disputes about the nature of mathematical objects, mathematical knowledge, etc. have shaped its present form and, moreover, continue to do so.
At a first glance, what Gödel was arguing for can be understood as the possibility that the concept of set will imply new essential properties that we could recognize as being new axioms of set theory. The task of the set theorist/phenomenologist would be, then, to: (i) provide a phenomenological account of the current state of set theory (say, of the current ZFC axioms) and (ii) show how this account can be extended in order to vindicate new axioms. Despite this, one can see the matter differently: with others like Dummett, Gödel would be recongnizing a feature of the notion of set itself, namely, its ‘indefinite extensibility’. This would be a candidate for new axiom itself, rather than other more convoluted properties.
Additionally, Gödel also adopted the ‘absoluteness principle’, i.e. that V is Absolute. Suppose that the notion of set did not satisfy the indefinite extensibility principle; then, this would mean that, at some point, such notion would stop implying new evident properties, etc. Hence, as a result, we would in some sense domesticate V, against the absoluteness principle. In other words: the Absolute conception entails the extensibility principle; being stuck in a closed formal specification is something that the notion of set cannot verify due to its absoluteness.
What struck me of this reasoning is that, despite not entering in technical details of the particular notion of set, it allowed a discussion on the nature of indefinite extensible concepts. This insight, coupled with Gödel’s purported theological argument, led me to asking for some general principles that the notion of indefinite extensibility satisfied >as I have read later, it seems that Dummett did a similar thing<. More precisely, since Gödel’s original assessment of CH made use of the notion of ‘intrinsic properties’ of set (against ‘extrinsic’ ones), I committed to the idea of developing such principles governing this concept.