  One would have thought that that was that, but Gentzen had reason to  produce even a fourth proof of the consistency of arithmetic, in his  last paper published in 1943 but written before the war in 1939. He  extended Peano arithmetic through transfinite ordinals and made the  transfinite induction principle part of this extended calculus. Then  he showed directly that transfinite induction up to the first  epsilon-number ε0 is expressible but not provable  in the system. Gödel's incompleteness theorem is thus proved in  a completely different way. The idea of the proof is, in brief terms, as follows: first it is laid down what it means to derive transfinite induction to a specific ordinal number in the system. Secondly,  ordinal numbers below ε0 are associated to  derivations. These are called “values”. It is then shown  that if transfinite induction to an ordinal number is derivable, this ordinal number cannot be greater than the value of the derivation.  Therefore transfinite induction to ε0 is not  derivable.