What precisely do you mean by non-standard set theory?
I didn’t say “non-standard set theory”. I said “non-standard models of set theory”.
I originally considered using “non-standard models of arithmetic” as my example of a fundamental, but unimportant question, but rejected it because the question is just too simple. Asking about non-standard models of set theory (models of ZFC, for example) is more comparable to the zombie question precisely because the question itself is less well defined. For example, just what do we mean in talking about a ‘model’ of ZFC, when ZFC or something similar is exactly the raw material used to construct models in other fields?
What axioms of set theory one is using can be important, and thinking about alternative models of set theory can lead to practical results.
Oh, I agree that some (many?) mathematicians will read Aczel (I didn’t realize the book was available online. Thx) and Barwise on AFA, and that even amateurs like me sometimes read Nelson, Steele, or Woodin. Just as AI researchers sometimes read Chalmers.
My point is that the zombie question may be interesting to an AI researcher, just as inaccessible cardinals or non-well-founded sets are interesting to an applied mathematician. But they are not particularly useful to most of the people who find them interesting. Most of the applications that Barwise suggests for Aczel’s work can be modeled with just a little more effort in standard ZF or ZFC. And I just can’t imagine that an AI researcher will learn anything from the p-zombie debate which will tell him which features or mechanisms his AI must have so as to avoid the curse of zombiedom.
For example, just what do we mean in talking about a ‘model’ of ZFC, when ZFC or something similar is exactly the raw material used to construct models in other fields?
Learning to distinguish different levels of formalism by training to follow mathematical arguments from formal set theory can help you lots in disentangling conceptual hurdles in decision theory (in its capacity as foundational study of goal-aware AI). It’s not a historical accident I included these kinds of math in my reading list on FAI.
Hmmm. JoshuaZ made a similar point. Even though the subject matter and the math itself may not be directly applicable to the problems we are interested in, the study of that subject matter can be useful by providing exercise in careful and rigorous thinking, analogies, conceptual structures, and ‘tricks’ that may well be applicable to the problems we are interested in.
I can agree with that. At least regarding the topics in mathematical logic we have been discussing. I am less convinced of the usefulness of studying the philosophy of mind. That branch of philosophy still strikes me as just a bunch of guys stumbling around in the dark.
I am less convinced of the usefulness of studying the philosophy of mind. That branch of philosophy still strikes me as just a bunch of guys stumbling around in the dark.
And I agree. The way Eliezer refers to p-zombie arguments is to draw attention to a particular error in reasoning, an important error one should learn to correct.
Asking about non-standard models of ZFC is deeply connected to asking about ZFC with other axioms added. This is connected to the Löwenheim–Skolem theorem and related results. Note for example that if there is some large cardinal axiom L and statement S such that ZFC + L can model ZFC + S, and L is independent of ZFC, then ZFC + S is consistent if ZFC is.
For example, just what do we mean in talking about a ‘model’ of ZFC, when ZFC or something similar is exactly the raw material used to construct models in other fields?
We can make this precise by talking about any given set theory as your ground and then discussing the models in it. This is connected to Paul Cohen’s work in forcing but I don’t know anything about it in any detail. The upshot though is that we can talk about models in helpful ways.
But they are not particularly useful to most of the people who find them interesting. Most of the applications that Barwise suggests for Aczel’s work can be modeled with just a little more effort in standard ZF or ZFC. And I just can’t imagine that an AI researcher will learn anything from the p-zombie debate which will tell him which features or mechanisms his AI must have so as to avoid the curse of zombiedom.
Not much disagreement there, but I think you might underestimate the helpfulness of thinking about different base axioms rather than talking about things in ZFC. In any event, the objection is not to your characterization of thinking about p-zombie but rather the analogy. The central point you are making seems correct to me.
I didn’t say “non-standard set theory”. I said “non-standard models of set theory”.
I originally considered using “non-standard models of arithmetic” as my example of a fundamental, but unimportant question, but rejected it because the question is just too simple. Asking about non-standard models of set theory (models of ZFC, for example) is more comparable to the zombie question precisely because the question itself is less well defined. For example, just what do we mean in talking about a ‘model’ of ZFC, when ZFC or something similar is exactly the raw material used to construct models in other fields?
Oh, I agree that some (many?) mathematicians will read Aczel (I didn’t realize the book was available online. Thx) and Barwise on AFA, and that even amateurs like me sometimes read Nelson, Steele, or Woodin. Just as AI researchers sometimes read Chalmers.
My point is that the zombie question may be interesting to an AI researcher, just as inaccessible cardinals or non-well-founded sets are interesting to an applied mathematician. But they are not particularly useful to most of the people who find them interesting. Most of the applications that Barwise suggests for Aczel’s work can be modeled with just a little more effort in standard ZF or ZFC. And I just can’t imagine that an AI researcher will learn anything from the p-zombie debate which will tell him which features or mechanisms his AI must have so as to avoid the curse of zombiedom.
Learning to distinguish different levels of formalism by training to follow mathematical arguments from formal set theory can help you lots in disentangling conceptual hurdles in decision theory (in its capacity as foundational study of goal-aware AI). It’s not a historical accident I included these kinds of math in my reading list on FAI.
Hmmm. JoshuaZ made a similar point. Even though the subject matter and the math itself may not be directly applicable to the problems we are interested in, the study of that subject matter can be useful by providing exercise in careful and rigorous thinking, analogies, conceptual structures, and ‘tricks’ that may well be applicable to the problems we are interested in.
I can agree with that. At least regarding the topics in mathematical logic we have been discussing. I am less convinced of the usefulness of studying the philosophy of mind. That branch of philosophy still strikes me as just a bunch of guys stumbling around in the dark.
And I agree. The way Eliezer refers to p-zombie arguments is to draw attention to a particular error in reasoning, an important error one should learn to correct.
Asking about non-standard models of ZFC is deeply connected to asking about ZFC with other axioms added. This is connected to the Löwenheim–Skolem theorem and related results. Note for example that if there is some large cardinal axiom L and statement S such that ZFC + L can model ZFC + S, and L is independent of ZFC, then ZFC + S is consistent if ZFC is.
We can make this precise by talking about any given set theory as your ground and then discussing the models in it. This is connected to Paul Cohen’s work in forcing but I don’t know anything about it in any detail. The upshot though is that we can talk about models in helpful ways.
Not much disagreement there, but I think you might underestimate the helpfulness of thinking about different base axioms rather than talking about things in ZFC. In any event, the objection is not to your characterization of thinking about p-zombie but rather the analogy. The central point you are making seems correct to me.