Definability of Goedel’s pairing function on ordinals

Given an infinite cardinal $kappa$, Goedel’s function is a well-known bijection $p:kappa^2$ onto $kappa$.

Is $p$ definable in the structure $<kappa;in>$?

Is $p$ definable in a bigger 2nd order structure $<kappa;mathcal P(kappa);in>$?

It looks like any typical attempt to code something like this (even + on ordinals) somehow refers to a pairing function.