ZT5's Shortform

The Parable of Gödel-Löb.

Three people discovered that no formal system can prove its own validity [LW · GW].

The first person decided to trust his own reasoning unconditionally. This allowed him to declare anything he wished as true, as he had already chosen to see all of his conclusions as correct. He declared he could fly, and jumped off a cliff. Reality disagreed.

The second person decided to mistrust his own reasoning unconditionally. He could not formally prove that he needed to eat and drink to survive, so he laid down on the ground and did nothing until he died of starvation.

The third person did something clever, yet obvious, and decided to trust his reasoning in a safe and bounded manner, and also mistrust his reasoning in a safe and bounded manner. He went on to explore reality, and was proven wrong about many of his conclusions (as choosing to trust his reasoning did not automatically make it correct). But ultimately he believed he made the correct choice, as taking the first step to greater understanding required him to have some degree of trust in himself.

It's weird how in common language use saying something is provable is a stronger statement than merely saying it is true.
While in mathematical logic it is a weaker statement.
Because truth already implies provability.
But provability doesn't imply truth.

My intuition felt it was important that I understand Löb's theorem. Because that's how formal logic systems decide to trust themselves, despite not being able to fully prove themselves to be valid (due to Gödel's incompleteness theorem). Which applies to my own mind as well.
This has always been true. But now I know. Updated on this.