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Asaf Karagila ♦
Asaf Karagila ♦
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We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. This technique is used, for example, in the proof that GBC is a conservative extension of ZFC, by forcing to add a universal choice function to a model of ZFC.

I'd like to know whether every class that can be 'safely' added to a model (preserving GBC and adding no sets) arises through forcing:

If $M$ is a model of GBC and $G$ is a class (that is, a subcollection of the sets of $M$, not a class member of $M$) such that $M[G]$ satisfies GBC and has the same sets as $M$, then is $G$ necessarily generic for some partial order $P \in M$?

I feel 'morally certain' that the answer must be no. Certainly the analogous question about sets, 'Can every set be added by forcing?' has a negative answer - we cannot, for example, force over $L$ to add a measure to a cardinal $\kappa$ (or to add $0^\#$$0^\\#$, or any other set that might increase the consistency strength). However, I don't see how to adapt this kind of argument to classes, if we are not allowed to add sets. I'd love to see a counterexample (a proof of a positive answer would also be welcome!).

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. This technique is used, for example, in the proof that GBC is a conservative extension of ZFC, by forcing to add a universal choice function to a model of ZFC.

I'd like to know whether every class that can be 'safely' added to a model (preserving GBC and adding no sets) arises through forcing:

If $M$ is a model of GBC and $G$ is a class (that is, a subcollection of the sets of $M$, not a class member of $M$) such that $M[G]$ satisfies GBC and has the same sets as $M$, then is $G$ necessarily generic for some partial order $P \in M$?

I feel 'morally certain' that the answer must be no. Certainly the analogous question about sets, 'Can every set be added by forcing?' has a negative answer - we cannot, for example, force over $L$ to add a measure to a cardinal $\kappa$ (or to add $0^\#$, or any other set that might increase the consistency strength). However, I don't see how to adapt this kind of argument to classes, if we are not allowed to add sets. I'd love to see a counterexample (a proof of a positive answer would also be welcome!).

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. This technique is used, for example, in the proof that GBC is a conservative extension of ZFC, by forcing to add a universal choice function to a model of ZFC.

I'd like to know whether every class that can be 'safely' added to a model (preserving GBC and adding no sets) arises through forcing:

If $M$ is a model of GBC and $G$ is a class (that is, a subcollection of the sets of $M$, not a class member of $M$) such that $M[G]$ satisfies GBC and has the same sets as $M$, then is $G$ necessarily generic for some partial order $P \in M$?

I feel 'morally certain' that the answer must be no. Certainly the analogous question about sets, 'Can every set be added by forcing?' has a negative answer - we cannot, for example, force over $L$ to add a measure to a cardinal $\kappa$ (or to add $0^\\#$, or any other set that might increase the consistency strength). However, I don't see how to adapt this kind of argument to classes, if we are not allowed to add sets. I'd love to see a counterexample (a proof of a positive answer would also be welcome!).

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jonasreitz
jonasreitz
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Is every class that does not add sets necessarily added by forcing?

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. This technique is used, for example, in the proof that GBC is a conservative extension of ZFC, by forcing to add a universal choice function to a model of ZFC.

I'd like to know whether every class that can be 'safely' added to a model (preserving GBC and adding no sets) arises through forcing:

If $M$ is a model of GBC and $G$ is a class (that is, a subcollection of the sets of $M$, not a class member of $M$) such that $M[G]$ satisfies GBC and has the same sets as $M$, then is $G$ necessarily generic for some partial order $P \in M$?

I feel 'morally certain' that the answer must be no. Certainly the analogous question about sets, 'Can every set be added by forcing?' has a negative answer - we cannot, for example, force over $L$ to add a measure to a cardinal $\kappa$ (or to add $0^\#$, or any other set that might increase the consistency strength). However, I don't see how to adapt this kind of argument to classes, if we are not allowed to add sets. I'd love to see a counterexample (a proof of a positive answer would also be welcome!).