> If there were, then a TM could follow that algorithm to construct its own Godel sentence. I don't understand what's the problem here. TMs are allowed to be i

> If there were, then a TM could follow that algorithm to construct its own Godel sentence. I don't understand what's the problem here. TMs are allowed to be inconsistent. There's a TM that outputs every single finite string. What would even count as its Godel sentence? In fact, there is an algorithm that constructs the Godel sentence given a consistent formal system capable of arithmetic. That algorithm is spelled out in Godel's original proof. Now, of course, a TM can only do that to its own formal system if it's inconsistent itself, which means the "Godel sentence" it output is meaningless.