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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.

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    $\begingroup$ The second sentence certainly does not mean anything. It is not what large cardinals are for. The answer to the question "Is this right?" is no, definitely. There are uses: Combinatorially they have rich properties. Metamathematically, they allow us to calibrate the consistency strength of theories extending $\mathsf{ZF}$. Mathematically, they provide us with a coherent picture of the theory of the reals beyond that given by the standard axioms, a theory that is complete in a profound technical sense. $\endgroup$ – Andrés E. Caicedo Jun 6 '14 at 5:24
  • $\begingroup$ @caicedo: sure, they're useful; I'm not disputing that; I'm asking why are they axioms; Hilbert spaces are useful, and there are axioms that define them, but they're not at foundational level. Are you saying that large cardinal axioms are not foundational? $\endgroup$ – Mozibur Ullah Jun 6 '14 at 5:32
  • $\begingroup$ @Caicedo: I've rephrased the question heading to make this more explicit. $\endgroup$ – Mozibur Ullah Jun 6 '14 at 5:47
  • $\begingroup$ The new title is completely different. Still, the body of the question talks of something else. If you edit your first comment into the question, that would go a long way into clarifying what you are after. $\endgroup$ – Andrés E. Caicedo Jun 6 '14 at 5:47
  • $\begingroup$ @Caicedo: done. $\endgroup$ – Mozibur Ullah Jun 6 '14 at 5:57
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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,

There exists a measurable cardinal.

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.

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