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Monroe Eskew
Monroe Eskew
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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 ZFCZF-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.

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 ZFC-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.

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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Monroe Eskew
Monroe Eskew
  • 18.6k
  • 5
  • 53
  • 115

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 ZFC-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.