This is a short post that offers a slightly different take on the standard proof of Löb's theorem. It offers nothing else of any value :)
We seek to prove the "inner" version, which we write as:
□P↔□(□P→P)
The proof uses quining to build a related sentence L, the "Löb sentence", which talks about its own source code. By construction L has the property:
□L↔□(□L→P)
Then, we can show that □L↔□P, i.e. they're equivalent! We do this by plugging □L into itself to get a twisty □P. We can then replace each □L with □P and prove Löb's theorem.
The proof
This proof uses the same rules of box manipulation as on the wiki page. We start by creating... (read 336 more words →)
What interesting things do y'all think are up with AI lab politics these days? Also why is everyone (or just many people in these circle) going to Anthropic now?
Any changes in how things seem for control plans based on vibes and awareness present in more recent models? (GPT-5 series may not count here; I'm mostly interested in visiblity on the next generation that are coming, of which I think Opus 4.5 is a preview but I'm fairly unsure.)
Anything generally striking about how things look in the landscape and models versus a year ago?