User blog comment:Meowzz/Question: 0=1 Cardinal/@comment-27513631-20180422220625

In a very generalised way, there are 'largest cardinal schemas' in other philosophies, typically incompatible with ZFC. For instance, predicitavism and finitism both have (small) ordinal bounds, so there could be seen to be viewpoints with a 'strongest axiom'. Rathjen has done works on upper-bounding constructivism as \(\Pi_1^2-\text{CA}\), I believe.

However, due to how unprincipled ZFC is, I doubt that this will occur there in a definite sense (there's no justification for what is/isn't a valid extension, except consistency).