Gentzen's Consistency Proof confusion I am recently finding some confusion. Some texts say that Gentzen's Consistency Proof shows transfinite induction up to $\varepsilon_0$ holds, while other texts say that consistency can be shown up to the numbers less than $\varepsilon_0$, but not $\varepsilon_0$. Which one is correct?
Thanks. 

Since $\epsilon_0$ is a limit ordinal when you say induction up to $\epsilon_0$ you mean every ordinal $<\epsilon_0$. In fact the confusion is only understanding in the terminology used, as both mean the same thing.
For example, induction on all the countable ordinals would be just the same as induction up to $\omega_1$.
