To convey my argument without as much of the math:
Suppose that P(X) is 1⁄2 at some stage. Then there will be inconsistent sets yet to remove which will take it at least C away from 1⁄2, where C is a constant that does not go down as the process continues.
The intuition behind this is that removing an inconsistent sentence which has not appeared in any of the inconsistencies removed so far, reduces mass by 1⁄2. Thus, the mass is changing significantly, all the time. Now to make this into a proof we need to show that P(X) changes significantly no matter how far into the process we go; IE, we need to show that a significantly different amount of mass can be removed from the ‘top’ and the ‘bottom’ (in the fraction M(X) / M(all) at finite depth).
The inconsistency {Y, Y->X, ~X} is supposed to achieve this: it only removes mass from the bottom, but there are infinitely many sets like this (we can make Y arbitrarily complex), and each of them reduces the bottom portion by the same fraction without touching the top. Specifically, the bottom becomes 7/8ths of its size (if I’ve done it right), so P(x) becomes roughly .57.
The fraction can re-adjust by decreasing the top in some other way, but this doesn’t allow convergence. No matter how many times the fraction reaches .5 again, there’s a new Y which can be used to force it to .57.
To convey my argument without as much of the math:
Suppose that P(X) is 1⁄2 at some stage. Then there will be inconsistent sets yet to remove which will take it at least C away from 1⁄2, where C is a constant that does not go down as the process continues.
The intuition behind this is that removing an inconsistent sentence which has not appeared in any of the inconsistencies removed so far, reduces mass by 1⁄2. Thus, the mass is changing significantly, all the time. Now to make this into a proof we need to show that P(X) changes significantly no matter how far into the process we go; IE, we need to show that a significantly different amount of mass can be removed from the ‘top’ and the ‘bottom’ (in the fraction M(X) / M(all) at finite depth).
The inconsistency {Y, Y->X, ~X} is supposed to achieve this: it only removes mass from the bottom, but there are infinitely many sets like this (we can make Y arbitrarily complex), and each of them reduces the bottom portion by the same fraction without touching the top. Specifically, the bottom becomes 7/8ths of its size (if I’ve done it right), so P(x) becomes roughly .57.
The fraction can re-adjust by decreasing the top in some other way, but this doesn’t allow convergence. No matter how many times the fraction reaches .5 again, there’s a new Y which can be used to force it to .57.