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.
A Hilbert space is a set with some structure. It's not hard to show in ZFC that Hilbert spaces exist, but if we wanted to we could consider some extremely weak theory $T\subset ZFC$ which can't prove e.g. that the reals form a set; then the statement "There is a Hilbert space" might not be provable in $T$, and would - if we wanted to use Hilbert spaces - need to be added as an axiom.
Large cardinal axioms are this, but in a context that's less silly. A large cardinal is a set with some properties (for various reasons, I hesitate to say "structure" here); for instance, a measurable is a cardinal which carries a countably complete ultrafilter (or many other equivalent formulations). The large cardinal axiom for a measurable is the statement,
This is not the same sort of thing as the defining "axioms" for a Hilbert space; those are analogous instead to the statement "$\kappa$ is measurable iff $\kappa$ is a cardinal and there is a countably complete ultrafilter on $\kappa$ which is nonprincipal;" both are really definitions. But we can make sense of a definition without knowing that something satisfying it exists! So this is the sense in which an axiom is required: without the large cardinal axioms, we can't tell that the large cardinal definitions apply to anything.