Are all large cardinal axioms expressible in terms of elementary embeddings? An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.
A critical point of such an embedding is the least ordinal $\alpha$ such that $f(\alpha)\neq\alpha$.  
A large cardinal is a cardinal number that cannot be proven to exist within ZFC.
They often appear to be critical points of an elementary embedding of models of ZFC where $M$ is the von Neumann hierarchy, and $N$ is some transitive model. Is this in fact true for all large cardinal axioms?

There is an online PDF of lecture slides by Woodin on the "Omega Conjecture" in which he axiomatizes the type of formulas that are large cardinals.  I do not know how exhaustive his formulation is. See the references under
http://en.wikipedia.org/wiki/Omega_conjecture
