# 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 $$$$?

Is $$p$$ definable in a bigger 2nd order structure $$$$?

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