I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of irreducible representations of $\mathcal W_F$.

It seems to me that strictly speaking this does not exist (due to set-theoretical reasons). Instead one thinks of $\hat{\mathcal{W}}_F$ as a set that contains for every irreducible representation of $\mathcal{W}_F$ an isomorphic copy of it. So elements of $\hat{\mathcal{W}}_F$ are true representations (and not some collection of representations). In order to create $\hat{\mathcal{W}}_F$ we need to be able to make a universal choice to pick a 'canonical model' from every class of isomorphic irreducible representations.

My questions are the following:

- Which axioms people assume in order to define a true set like $\hat{\mathcal{W}}_F$?
- Does it following from the axioms used for example in Bourbaki? (to answer this one needs to be familiar with their foundations as described in 'Theory of Sets'.

Other considerations/remarks related to my question are also welcome. Thanks in advance.

oneuniversal choice from every class, you can make a nonempty set of choices from each class. Scott's Trick is a systematic way of doing this in ZF and in Bourbaki's system: en.wikipedia.org/wiki/Scott%27s_trick $\endgroup$ – François G. Dorais♦ Aug 15 '15 at 12:47setof equivalence classes), because you can just bound the cardinality of all the irreducible representations. $\endgroup$ – Eric Wofsey Aug 15 '15 at 12:51