A cardinal is an isomorphism class in ZFC, or a representative of one.
I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm asking why are they axioms; Hilbert spaces are useful, and there are axioms that define them, but they're not at the (set-theoretical) foundational level. Can we say that large cardinal axioms are not foundational?
My understanding is that they are in fact foundational (going by wikipedia); is this correct?
Further, (again going by wikipedia) large cardinals axioms are necessary when we can't show that certain cardinals exist; and I understand by this that some condition is given that picks out a cardinality. But we can't show that we can construct that set by the operations of set-theory starting from any previously constructed cardinals (the first one is a given by the axiom of infinity).
Is this right?
I apologise for the lack of precision in this question - it was going to be a question on Philosophy.SE; but then I thought this site would be better.