Universality has various interpretations and it will be the purpose of this talk to establish that many of these are provably different. A graph can be thought of as a function from pairs to 2. It was established by Shelah that it is consistent with the failure of CH that there is a graph of cardinality \aleph_1 that is universal for graphs of cardinality \aleph_1; in other words, there is a function from pairs of \omega_1 to 2 which embeds any other such function. One can use Shelah's ideas to establish that it is consistent with the failure of CH that there is a function from pairs of \omega_1 to \omega that is universal for functions of cardinality \aleph_1. It will be shown in this talk that it is consistent that there is a universal function into 2 but not into \omega. This is joint work with Shelah.
It will also be shown how to modify this argument to show that it is consistent there is no function from pairs of \omega_1 to \omega that is universal for functions of cardinality \aleph_1 yet there is such a function of \omega is treated as a set of indiscernibles.