Let P be your statement. Then P = I can't prove P.

For the purpose of argument, let's make the bold assumption that I can prove any statement that can be proven under a given consistent system. (By the way, we shall assume the system we're using is consistent.)

Then P = ~provable(P). So ~P = provable(P).

Which indicates, if provable(P), then ~P. But if ~P, then obviously ~provable(P). Therefore, by transitivity of implication, if provable(P), then ~provable(P).

Any statement that implies its own inverse is merely a special case of proof by contradiction, whereby premises leading to a contradictory conclusion must be rejected. Therefore, if (if provable(P) then ~provable(P)) then ~provable(P). But we've already proved that if provable(P), then ~provable(P).

Therefore, ~provable(P).

But we've already established P = ~provable(P). Thus, from the premise P = ~provable(P), I have proven P to be true.

HOWEVER: if I have proven P to be true, under no premises but its own definition, then provable(P). But we have already established ~provable(P).

Again we have reached a contradiction, and are forced to conclude one of our premises is false. But my only formal premise in this logical argument was that P = ~provable(P).

Therefore, P does not equal ~provable(P). In other words, the statement you've submitted is logically invalid in form.

Prove it..

Let P be your statement. Then P = I can't prove P.

For the purpose of argument, let's make the bold assumption that I can prove any statement that can be proven under a given consistent system. (By the way, we shall assume the system we're using is consistent.)

Then P = ~provable(P). So ~P = provable(P).

Which indicates, if provable(P), then ~P. But if ~P, then obviously ~provable(P). Therefore, by transitivity of implication, if provable(P), then ~provable(P).

Any statement that implies its own inverse is merely a special case of proof by contradiction, whereby premises leading to a contradictory conclusion must be rejected. Therefore, if (if provable(P) then ~provable(P)) then ~provable(P). But we've already proved that if provable(P), then ~provable(P).

Therefore, ~provable(P).

But we've already established P = ~provable(P). Thus, from the premise P = ~provable(P), I have proven P to be true.

HOWEVER: if I have proven P to be true, under no premises but its own definition, then provable(P). But we have already established ~provable(P).

Again we have reached a contradiction, and are forced to conclude one of our premises is false. But my only formal premise in this logical argument was that P = ~provable(P).

Therefore, P does not equal ~provable(P). In other words, the statement you've submitted is logically invalid in form.