How do you interpret “soundness”? It’s being used to mean that a proof of X implies X, for any statement X in the formal language of the theory. And yes, Löb’s Theorem directly shows that PA cannot prove its own soundness for any set of statements save a subset of its own theorems.
I consider myself to be a reasoner at least as powerful as PA, since I am using myself plus PA plus Löb’s Theorem to reason about systems at least as powerful as myself (which encompasses everything I’ve thus far described).
I consider myself “sound” if I believe myself to be trustworthy / reliable enough to believe what I believe Löb’s Theorem’s says, which is something to do with self-trust and the ability to believe that certain propositions are true, as long as I can prove that proving them true means that they are true.
Do you also believe that if you could “prove” eating babies was morally required, eating babies would be morally required? PA obviously believes Lob’s theorem itself, and indeed proves the soundness of all its actual proofs, which is what I said above. What PA doesn’t trust is hypothetical proofs.
I don’t know that I follow. The question, here and in the context of Löb’s Theorem, is about hypothetical proofs. Do you trust yourself enough to say that, in the hypothetical where you experience a proof that eating babies is mandatory, that would mean eating babies is mandatory?
“Experiencing a proof that eating babies is mandatory” could only amount to something like being force-fed or otherwise some kind of impossible to resist scenario. “Experiencing a proof” of any proposition consists of the sequence of events or observed inputs leading to one to conclude a proposition is true.
If you are firmly in the against baby eating camp, then presumably you’ve got ample proof already that it is not mandatory, and always generally possible to resist anyone trying to make you eat them.
I think you’re talking about the hypothetical world in which both baby eating and not baby eating seem correct, such as where one hypothetically reads a proof that seems correct, and urges one to eat babies, whilst it simultaneously seeming very wrong according to our normal intuitions like it normally does. That hypothetical world is inconsistent by assumption, though. I’m not talking about that. I think we live in the world in which our sense that baby eating is very wrong means a proof of that can be constructed.
How do you interpret “soundness”? It’s being used to mean that a proof of X implies X, for any statement X in the formal language of the theory. And yes, Löb’s Theorem directly shows that PA cannot prove its own soundness for any set of statements save a subset of its own theorems.
I consider myself to be a reasoner at least as powerful as PA, since I am using myself plus PA plus Löb’s Theorem to reason about systems at least as powerful as myself (which encompasses everything I’ve thus far described).
I consider myself “sound” if I believe myself to be trustworthy / reliable enough to believe what I believe Löb’s Theorem’s says, which is something to do with self-trust and the ability to believe that certain propositions are true, as long as I can prove that proving them true means that they are true.
Do you also believe that if you could “prove” eating babies was morally required, eating babies would be morally required? PA obviously believes Lob’s theorem itself, and indeed proves the soundness of all its actual proofs, which is what I said above. What PA doesn’t trust is hypothetical proofs.
No, I would have to think that eating babies was morally required first. If not, then I could not prove it.
I don’t know that I follow. The question, here and in the context of Löb’s Theorem, is about hypothetical proofs. Do you trust yourself enough to say that, in the hypothetical where you experience a proof that eating babies is mandatory, that would mean eating babies is mandatory?
“Experiencing a proof that eating babies is mandatory” could only amount to something like being force-fed or otherwise some kind of impossible to resist scenario. “Experiencing a proof” of any proposition consists of the sequence of events or observed inputs leading to one to conclude a proposition is true.
If you are firmly in the against baby eating camp, then presumably you’ve got ample proof already that it is not mandatory, and always generally possible to resist anyone trying to make you eat them.
I think you’re talking about the hypothetical world in which both baby eating and not baby eating seem correct, such as where one hypothetically reads a proof that seems correct, and urges one to eat babies, whilst it simultaneously seeming very wrong according to our normal intuitions like it normally does. That hypothetical world is inconsistent by assumption, though. I’m not talking about that. I think we live in the world in which our sense that baby eating is very wrong means a proof of that can be constructed.