Large cardinals and $V$ I am confused by something: $\mu$ is a large cardinal if $\lambda<\mu\Rightarrow 2^{\lambda}<\mu$ and any union of less than $\mu$ sets of size less than $\mu$ is less than $\mu$. On Wikipedia it says "large cardinals are understood in the context of the von Neumann universe $V$". But isn't $V$ constructed precisely using the power set and union operations? 
I know we cannot prove the existence of large cardinals in $\sf ZFC$, but if we adopted "a large cardinal exists in $V$" why doesn't this create a contradiction? How can something exist in $V$ without being reachable using the tools $V$ is built from?
 A: This is a good question, and a delicate point.
The von Neumann hierarchy does not construct the universe. The universe is given to us, by the gods, by ourselves, by whoever wrote the text. What is true, however, is that given a universe of $\sf ZF$ then we can prove that that von Neumann hierarchy is a hierarchy which covers the entire universe. That is, every set in $V$ is in some $V_\alpha$.
Ask yourself, if the von Neumann hierarchy actually constructs the universe, then where do the ordinals come from? Why can we continue after reaching ordinals like the first $\aleph$ fixed point? Or the first $\beth$ fixed point? Or the first ordinal which is both an $\aleph$ and a $\beth$ fixed point?
No. The universe is given.
We can, however, stretch this slightly more. Suppose that we have a universe satisfying $\sf ZF-Fnd$, then the von Neumann hierarchy constructs an inner model $V$ which satisfies $\sf ZF$. Why can we say construct in this context? Because we construct an inner model, but this means that a universe is given to us already.
So if $\kappa$ is an inaccessible cardinal, it already exists in the universe. The point is that the first step any set of cardinality $\kappa$ appears in the von Neumann construction is $\kappa$. It doesn't imply any contradiction, much like how the existence of an inaccessible cardinal implies the consistency of $\sf ZFC$ doesn't imply a contradiction either.
And just to finish with a small point, large cardinals don't always have to be inaccessible. See about Rowbottom and Jonsson cardinals. Or even actual cardinals, see about $0^\#$ (read: zero sharp) which is in fact a real number.
