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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?

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  • $\begingroup$ In first order set theory, proper classes do no exist as objects. We can make the notion of "internal property" precise by identifying it with first-order property. $\endgroup$
    – Monroe Eskew
    Commented Oct 5, 2018 at 13:40
  • $\begingroup$ @MonroeEskew The notion of an internal property is supposed to extend to logics of any order (cf. Godel's original characterization in Wang 1996). $\endgroup$
    – Mallik
    Commented Oct 5, 2018 at 15:01
  • $\begingroup$ I was only suggesting one way to go. What do you mean by "including proper classes in our domain"? It is consistent relative to large cardinals that the universe reflects 2nd order logic. $\endgroup$
    – Monroe Eskew
    Commented Oct 5, 2018 at 15:11
  • $\begingroup$ @MonroeEskew Thanks for the question, I was imprecise. I've edited my question to reflect better what I had in mind, hopefully. $\endgroup$
    – Mallik
    Commented Oct 6, 2018 at 4:08

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Referring to proper classes by means of constants clearly doesn’t use the “internal properties” of the universe of sets. You’re literally reaching outside the universe and adding an artificial means of referring to a class.

Regarding (1), I think the intuition is that classes are not objects of the model, but rather predicates. Viewed this way, even nondefinable classes cannot be forced to be contained in every elementary submodel, because the model simply has to agree with the predicate on its elements.

Regarding (2), again proper classes are not objects, but predicates. When we are talking about definable proper classes, then the ZF-provable Reflection Theorem captures the ineffability nicely: For whatever class you define by means of a first order formula with set parameters, whatever you want to say about the class, there will be some rank that interprets the class correctly, and agrees with $V$ about whatever you’ve claimed of it.

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