Obviously I’m aware of this. As stated several times, my original comment refers to the hypothetical situation in which PA could prove its consistency without Godel’s theorems being true/known. One would not be able to conclude anything.
The point being, having PA prove its own consistency couldn’t tell you anything of value even in the case that Godel’s theorems were not true. This is an interesting phenomenon. The only way to know a system is consistent is to know all of its theorems.
The point being, having PA prove its own consistency couldn’t tell you anything of value even in the case that Godel’s theorems were not true. This is an interesting phenomenon. The only way to know a system is consistent is to know all of its theorems.