error of using Löb theorem Under the assumption of consistency of a logic theory T, 
Logically, 

any statement follows from a contradiction sentence. 

For example,
"4 is odd number → 3 is even number"
0=1 → 3=4
Similarly,

(0=1 is provable) → (0=1)

For a logic theory T,
Löb theorem says that if T ⊢ { (P is provable) → P } ,  then T ⊢ P, so,

T ⊢ 0=1 . 

What's the error of this reasoning?
 A: Let's break down what Löb's theorem says, exactly, in this case. Löb's theorem asserts the following about some class of theories T:

If T proves that (if T proves P then P), then T proves P.

What does this say when P is a contradiction? Well, first let's break down the meaning of each part carefully. 

T proves P

means "T is inconsistent." Next,

(if T proves P then P)

means "if T is inconsistent, we get a contradiction," so it means "T is consistent." Next,

If T proves that (if T proves P then P)

means "if T proves that T is consistent." Finally,

If T proves that (if T proves P then P), then T proves P.

means "if T proves that T is consistent, then T is inconsistent." And this is precisely Gödel's second incompleteness theorem!
A: Löb's theorem says that if it is provable that P's provability implies P, then P is provable.  The statement "(0=1 is provable)→ (0=1)" is provably equivalent to " ¬(0=1 is provable)", so, by Gödel's second incompleteness theorem, it cannot be proved (assuming the system is consistent.)
