Actually, the `proof’ you gave that no true list of theories like this exists made the assumption (not listed in this paper) that the sequence of indexes for the computable theories is definable over arithmetic. In general there is no reason this must be true but of course for the purposes of an AI it must.
(“This paper” being Eliezer’s writeup of the procrastination paradox.) That’s true, thanks.
Ultimately, you can always collapse any computable sequence of computable theories (necessary for the AI to even manipulate) into a single computable theory so there was never any hope this kind of sequence could be useful.
First of all (always assuming the theories are at least as strong as PA), note that in any such sequence, T_0 is the union of all the theories in the sequence; if T_(n+1) |- phi, then PA |- Box_(T_(n+1)) “phi”, so T_n |- Box_(T_(n+1)) “phi”, so by the trust schema, T_n |- phi; going up the chain like this, T_0 |- phi. So T_0 is in fact the “collapse” of the sequence into a single theory.
That said, I disagree that there is no hope that this kind of sequence could be useful. (I don’t literally want to use an unsound theory, but see my writeup about an infinite sequence of sound theories each proving the next consistent, linked from the main post; the same remarks apply there.) Yes, T_0 is stronger than T_1, so why would you ever want to use T_1? Well, T_0 + Con(T_0) is stronger than T_0, so why would you ever want to use T_0? But by this argument, you can’t use any sound theory including PA, so this doesn’t seem like a remotely reasonable argument against using T_1. Moreover, the fact that an agent using T_0 can construct an agent using T_1, but it can’t construct an agent using T_0, seems like a sufficient argument against the claim that the sequence as a whole must be useless because you could always use T_0 for everything.
I meant useful in the context of AI since any such sequence would obviously have to be non-computable and thus not something the AI (or person) could make pragmatic use of.
Also, it is far from clear that T_0 is the union of all theories (and this is the problem in the proof in the other rightup). It may well be that there is a sequence of theories like this all true in the standard model of arithmetic but that their construction requires that Tn add extra statements beyond the schema for the proof predicate in T{n+1}
Also, the claim that Tn must be stronger than T{n+1} (prove a superset of it...to be computable we can’t take all these theories to be complete) is far from obvious if you don’t require that T_n be true in the standard model. If T_n is true in the standard model than, as it proves that Pf(Tn+1, \phi) → \phi this is true so if T{n+1} |- \phi then (as this witnessed in a finite proof) there is a proof that this holds from T_n and thus a proof of \phi. However, without this assumption I don’t even see how to prove the containment claim.
I meant useful in the context of AI since any such sequence would obviously have to be non-computable and thus not something the AI (or person) could make pragmatic use of.
I was replying to this:
Ultimately, you can always collapse any computable sequence of computable theories (necessary for the AI to even manipulate) into a single computable theory so there was never any hope this kind of sequence could be useful.
I.e., I was talking about computable sequences of computable theories, not about non-computable ones.
Also, it is far from clear that T_0 is the union of all theories (and this is the problem in the proof in the other rightup). It may well be that there is a sequence of theories like this all true in the standard model of arithmetic but that their construction requires that T_n add extra statements beyond the schema for the proof predicate in T_{n+1}
I can’t make sense of this. Of course T_n can contain statements other than those in T_{n+1} and the Löb schema of T_{n+1}, but this is no problem for the proof that T_0 is the union of all the theories; the point is that because of the Löb schema, we have T_{n+1} \subset T_n for all n, and therefore (by transitivity of the subset operation) T_n \subseteq T_0 for all n.
Also, the claim that T_n must be stronger than T_{n+1} (prove a superset of it...to be computable we can’t take all these theories to be complete) is far from obvious if you don’t require that T_n be true in the standard model. If T_n is true in the standard model than, as it proves that Pf(T_n+1, \phi) → \phi this is true so if T_{n+1} |- \phi then (as this witnessed in a finite proof) there is a proof that this holds from T_n and thus a proof of \phi. However, without this assumption I don’t even see how to prove the containment claim.
Note again that I was talking about computable sequences T_n. If T_{n+1} |- \phi and T_{n+1} is computable, then PA |- Pf(T_{n+1}, \phi) and therefore T_n |- Pf(T_{n+1}, \phi) if T_n extends PA. This doesn’t require either T_n or T_{n+1} to be sound.
(“This paper” being Eliezer’s writeup of the procrastination paradox.) That’s true, thanks.
First of all (always assuming the theories are at least as strong as PA), note that in any such sequence, T_0 is the union of all the theories in the sequence; if T_(n+1) |- phi, then PA |- Box_(T_(n+1)) “phi”, so T_n |- Box_(T_(n+1)) “phi”, so by the trust schema, T_n |- phi; going up the chain like this, T_0 |- phi. So T_0 is in fact the “collapse” of the sequence into a single theory.
That said, I disagree that there is no hope that this kind of sequence could be useful. (I don’t literally want to use an unsound theory, but see my writeup about an infinite sequence of sound theories each proving the next consistent, linked from the main post; the same remarks apply there.) Yes, T_0 is stronger than T_1, so why would you ever want to use T_1? Well, T_0 + Con(T_0) is stronger than T_0, so why would you ever want to use T_0? But by this argument, you can’t use any sound theory including PA, so this doesn’t seem like a remotely reasonable argument against using T_1. Moreover, the fact that an agent using T_0 can construct an agent using T_1, but it can’t construct an agent using T_0, seems like a sufficient argument against the claim that the sequence as a whole must be useless because you could always use T_0 for everything.
I meant useful in the context of AI since any such sequence would obviously have to be non-computable and thus not something the AI (or person) could make pragmatic use of.
Also, it is far from clear that T_0 is the union of all theories (and this is the problem in the proof in the other rightup). It may well be that there is a sequence of theories like this all true in the standard model of arithmetic but that their construction requires that Tn add extra statements beyond the schema for the proof predicate in T{n+1}
Also, the claim that Tn must be stronger than T{n+1} (prove a superset of it...to be computable we can’t take all these theories to be complete) is far from obvious if you don’t require that T_n be true in the standard model. If T_n is true in the standard model than, as it proves that Pf(Tn+1, \phi) → \phi this is true so if T{n+1} |- \phi then (as this witnessed in a finite proof) there is a proof that this holds from T_n and thus a proof of \phi. However, without this assumption I don’t even see how to prove the containment claim.
I was replying to this:
I.e., I was talking about computable sequences of computable theories, not about non-computable ones.
I can’t make sense of this. Of course T_n can contain statements other than those in T_{n+1} and the Löb schema of T_{n+1}, but this is no problem for the proof that T_0 is the union of all the theories; the point is that because of the Löb schema, we have T_{n+1} \subset T_n for all n, and therefore (by transitivity of the subset operation) T_n \subseteq T_0 for all n.
Note again that I was talking about computable sequences T_n. If T_{n+1} |- \phi and T_{n+1} is computable, then PA |- Pf(T_{n+1}, \phi) and therefore T_n |- Pf(T_{n+1}, \phi) if T_n extends PA. This doesn’t require either T_n or T_{n+1} to be sound.