can we a prove ⊢ (α → α) → (α → α) The system L0 is defined as follows:
Axioms:
A1 (α → (β → α))
A2 2. (α → (β → γ) → ((α → β) → (α → γ))
A3 ((¬β → ¬α) → ((¬β → α) → β))
In one of my problem sheets, I am told that I am allowed to use the following theorem:
if 
⊢ (α → α)  
⊢ (α → α) → (α → α)
My attempt to solve/prove the second one 
⊢ (α → α) → (α → α)
choose 2 axioms which match.
α →α 
β→α
γ→α
(α → (α → α) → ((α → α) → (α → α))
I'm struggling to see an obvious way of proving the theorem.
Is it provable? And if so, can you point me in the right direction?
Thanks a lot!

The other answers show how to prove $P\to P$ and then set $P\equiv(\alpha\to\alpha)$. This works, of course.
But your particular goal can be reached faster with the axioms you have:


*

*$(\alpha\to(\alpha\to\alpha))$

*$(\alpha\to(\alpha\to\alpha))\to((\alpha\to\alpha)\to(\alpha\to\alpha))$
are instaces of your axioms 1 and 2, respectively.
Now use Modus Ponens once and you're done.
