The real problem is trusting yourself because you trust yourself.
What it means to ‘trust’ PA is to say “PA can prove only things that are true”. It is impossible within PA to prove that, but it is not inherently impossible to prove that outside PA. I can believe that “PA can prove only things which are true” and use that as one of my premises without much trouble, because it is true. (If it isn’t, then PA+1 can prove anything)
The premise “I can prove only things which are true.” is what is falsified by Lob’s theorem- and it is falsified in the general case. What Lob’s theorem doesn’t prove is “No system can prove only things which are true.”
In reference to ethics and meta-ethics, PA makes no claims. What we need is a system that has proven as many things as possible, while proving no false things. In order to identify that, we need to figure out how to falsify a meta-ethical, ethical, or moral statement.
The real problem is trusting yourself because you trust yourself.
What it means to ‘trust’ PA is to say “PA can prove only things that are true”. It is impossible within PA to prove that, but it is not inherently impossible to prove that outside PA. I can believe that “PA can prove only things which are true” and use that as one of my premises without much trouble, because it is true. (If it isn’t, then PA+1 can prove anything)
The premise “I can prove only things which are true.” is what is falsified by Lob’s theorem- and it is falsified in the general case. What Lob’s theorem doesn’t prove is “No system can prove only things which are true.”
In reference to ethics and meta-ethics, PA makes no claims. What we need is a system that has proven as many things as possible, while proving no false things. In order to identify that, we need to figure out how to falsify a meta-ethical, ethical, or moral statement.