@abramdemski Wanted to say thanks again for engaging with my posts and pointing me towards looking again at Lob. It’s weird: now that I’ve taken so time to understand it, it’s just what in my mind was already the thing going on with Godel, just I wasn’t doing a great job of separating out what Godel proves and what the implications are. As presented on its own, Lob didn’t seem that interesting to me so I kept bouncing off it as something worth looking at, but now I realize it’s just the same thing I learned from GEB’s presentation of Peano arithmetic and Godel when I read it 20+ years ago.
When I go back to make revisions to the book, I’ll have to reconsider including Godel and Lob somehow in the text. I didn’t because I felt like it was a bit complicated and I didn’t really need to dig into it since I think there’s already a bit too many cases where people use Godel to overreach and draw conclusions that aren’t true, but it’s another way to explain these ideas. I just have to think about if Godel and Lob are necessary: that is, do I need to appeal to them to make my key points, or are these things that are better left as additional topics I can point folks at but not key to understanding the intuitions I want them to develop.
I’ve heard Lob remarked that he would never have published if he realized earlier how close his theorem was to just Godel’s second incompleteness theorem; but I can’t seem to entirely agree with Lob there. It does seem like a valuable statement of its own.
I agree, Godel is dangerously over-used, so the key question is whether it’s necessary here. Other formal analogs of your point include Tarski’s undefinability, and the realizablility / grain-of-truth problem. There are many ways to gesture towards a sense of “fundamental uncertainty”, so the question is: what statement of the thing do you want to make most central, and how do you want to argue/illustrate that statement?
@abramdemski Wanted to say thanks again for engaging with my posts and pointing me towards looking again at Lob. It’s weird: now that I’ve taken so time to understand it, it’s just what in my mind was already the thing going on with Godel, just I wasn’t doing a great job of separating out what Godel proves and what the implications are. As presented on its own, Lob didn’t seem that interesting to me so I kept bouncing off it as something worth looking at, but now I realize it’s just the same thing I learned from GEB’s presentation of Peano arithmetic and Godel when I read it 20+ years ago.
When I go back to make revisions to the book, I’ll have to reconsider including Godel and Lob somehow in the text. I didn’t because I felt like it was a bit complicated and I didn’t really need to dig into it since I think there’s already a bit too many cases where people use Godel to overreach and draw conclusions that aren’t true, but it’s another way to explain these ideas. I just have to think about if Godel and Lob are necessary: that is, do I need to appeal to them to make my key points, or are these things that are better left as additional topics I can point folks at but not key to understanding the intuitions I want them to develop.
I’ve heard Lob remarked that he would never have published if he realized earlier how close his theorem was to just Godel’s second incompleteness theorem; but I can’t seem to entirely agree with Lob there. It does seem like a valuable statement of its own.
I agree, Godel is dangerously over-used, so the key question is whether it’s necessary here. Other formal analogs of your point include Tarski’s undefinability, and the realizablility / grain-of-truth problem. There are many ways to gesture towards a sense of “fundamental uncertainty”, so the question is: what statement of the thing do you want to make most central, and how do you want to argue/illustrate that statement?