Strictly speaking, Lob’s Theorem doesn’t show that PA doesn’t prove that the provability of any statement implies that statement. It just shows that if you have a statement in PA of the form (If S is provable, then S), you can use this to prove S. The part about PA not proving any implications of that form for a false S only follows if we assume that PA is sound.
Therefore, replacing PA with a stronger system or adding primitive concepts of provability in place of PA’s complicated arithmetical construction won’t help. As long as it can do everything PA can do (for example, prove that it can prove things it can prove), it will always be able to get from (If S is provable, then S) to S, even if S is 3*5=56..
Let me see what happens if I put in a specific example. Suppose that
If 3*5=35 is a theorem of PA, then 3*5=35
is a theorem of PA. Let me refer to “3*5=35 is a theorem” as sentence 1; “3*5=35″ as sentence 2, and the implication 1->2 as sentence 3. Now, if 3 is a theorem, then you can use PA to prove 2 even without actually showing 1; and then PA has proven a falsehood, and is inconsistent. Is that a correct statement of the problem?
If so… I seem to have lost track of the original difficulty, sorry. Why is it a worry that PA will assert that it can prove something false, but not a worry that it will assert something false? If you’re going to worry that sentence 3 is a theorem, why not go straight to worrying that sentence 2 is a theorem?
We generally take it for granted that sentence 2 (because it is false) is not a theorem (and therefore sentence 1 is false), and the argument is meant to show that sentence 3 is therefore also not a theorem (even though it is true, since sentence 1 is false). This is a problem because we would like to use reasoning along the lines of sentence 3.
To see the real issue, you should replace sentence 2 with a conjecture whose truth you don’t yet know but would like to know (and modify sentences 1 and 3 correspondingly). Now wouldn’t you like sentence 3 to be a true? If you knew that sentence 1 was true, wouldn’t you like to conclude that sentence 2 is true? Yet if you’re PA, then you can’t do that.
Strictly speaking, Lob’s Theorem doesn’t show that PA doesn’t prove that the provability of any statement implies that statement. It just shows that if you have a statement in PA of the form (If S is provable, then S), you can use this to prove S. The part about PA not proving any implications of that form for a false S only follows if we assume that PA is sound.
Therefore, replacing PA with a stronger system or adding primitive concepts of provability in place of PA’s complicated arithmetical construction won’t help. As long as it can do everything PA can do (for example, prove that it can prove things it can prove), it will always be able to get from (If S is provable, then S) to S, even if S is 3*5=56..
Let me see what happens if I put in a specific example. Suppose that
is a theorem of PA. Let me refer to “3*5=35 is a theorem” as sentence 1; “3*5=35″ as sentence 2, and the implication 1->2 as sentence 3. Now, if 3 is a theorem, then you can use PA to prove 2 even without actually showing 1; and then PA has proven a falsehood, and is inconsistent. Is that a correct statement of the problem?
If so… I seem to have lost track of the original difficulty, sorry. Why is it a worry that PA will assert that it can prove something false, but not a worry that it will assert something false? If you’re going to worry that sentence 3 is a theorem, why not go straight to worrying that sentence 2 is a theorem?
We generally take it for granted that sentence 2 (because it is false) is not a theorem (and therefore sentence 1 is false), and the argument is meant to show that sentence 3 is therefore also not a theorem (even though it is true, since sentence 1 is false). This is a problem because we would like to use reasoning along the lines of sentence 3.
To see the real issue, you should replace sentence 2 with a conjecture whose truth you don’t yet know but would like to know (and modify sentences 1 and 3 correspondingly). Now wouldn’t you like sentence 3 to be a true? If you knew that sentence 1 was true, wouldn’t you like to conclude that sentence 2 is true? Yet if you’re PA, then you can’t do that.