A prominent idea found e.g. in Koellner (http://logic.harvard.edu/koellner/ORP_final.pdf) is that reflection principles in set theory are motivated by the idea that proper classes are so large as to be "ineffable" or unable to be characterized by any internal structural property of the universe (of which they are proper classes).

1. The notion of an "internal" property is vague. For example, what (part of the intuition) prevents simply including proper classes in our domain so that they fall within the range of our quantifiers, and are thereby characterized by "internal" properties involving universal quantification?

2. How does the supposed ineffability motivation fit with the existence of definable proper classes that (prima facie at least) seem to violate it?

A prominent idea found e.g. in Koellner (http://logic.harvard.edu/koellner/ORP_final.pdf) is that reflection principles in set theory are motivated by the idea that proper classes are so large as to be "ineffable" or unable to be characterized by any internal structural property of the universe (of which they are proper classes).

1. The notion of an "internal" property is vague. For example, suppose we add constants to our semantics that are stipulated to denote proper classes, and then construct formulas using these constants that are uniquely satisfied by those proper classes. Intuitively this ought to violate the ineffability of these classes (for we have referred to them by uniquely characterizing them), but if so, what makes the property in question an "internal" one?

2. How does the supposed ineffability motivation fit with the existence of definable proper classes that (prima facie at least) seem to violate it?