Eliezer: “Why doesn’t the given proof of Lob’s Theorem, the steps 1-10, show that PA + “◻C → C” |- C?”
That’s just not what it says. The hypothesis in step 2 isn’t “◻C → C” it’s “◻(◻C → C)”. I suppose if the Hypothesis were “◻C → C” we could try to find where it breaks. Step 7 is the only step that depends on 2 so it has to be there. Translating the amusing cartoons, we have
◻(◻C → C)
◻(◻L → ◻C)
◻(◻L → C)
Lets say that instead we have
2′. ◻C → C
Well what are we supposed to do with it? We’re stuck. Just because ◻C → C doesn’t mean that PA can prove it.
Eliezer: “Why doesn’t the given proof of Lob’s Theorem, the steps 1-10, show that PA + “◻C → C” |- C?”
That’s just not what it says. The hypothesis in step 2 isn’t “◻C → C” it’s “◻(◻C → C)”. I suppose if the Hypothesis were “◻C → C” we could try to find where it breaks. Step 7 is the only step that depends on 2 so it has to be there. Translating the amusing cartoons, we have
◻(◻C → C)
◻(◻L → ◻C)
◻(◻L → C)
Lets say that instead we have
2′. ◻C → C
Well what are we supposed to do with it? We’re stuck. Just because ◻C → C doesn’t mean that PA can prove it.