Prove that if $ \Gamma $ is inconsistent, then $\Gamma \vdash \beta $ for every formula $\beta$

Question

Considering that $\Gamma$ is inconsistent if $ \Gamma \vdash ¬(\alpha \rightarrow \alpha) $ for some formula $\alpha$.

How to prove that if $ \Gamma $ is inconsistent, then $ \Gamma \vdash \beta $?

Do I need to assume that a proof of $ ¬(\alpha \rightarrow \alpha) $ exists and try to find $\beta$ using axioms? Is it possible?

List of axioms:

Axioms

Answer 1

If you can prove $\Gamma \vdash ¬(\alpha \rightarrow \alpha)$, you can say $$\Gamma,\lnot \beta \vdash ¬(\alpha \rightarrow \alpha) \\ \Gamma \vdash (\lnot \beta \implies¬(\alpha \rightarrow \alpha)) \\ \Gamma \vdash \beta$$

Answer 2

Hint: Prove that, in Polish notation, $\vdash$CpCNpq.

More fully:

axiom 1 CpCqp
axiom 2 CCpCqrCCpqCpr
axiom 3 CCpqCCpNqNp
axiom 4 CNNpp
2 r/p 5 CCpCqpCCpqCpp
5, 1  6 CCpqCpp
6 q/Cqp 7 CCpCqpCpp
7, 1    8 Cpp
hypothesis  9 | p
hypothesis  10 || Np
1 q/Nq      11 || CpCNqp
1 q/Nq p/Np 12 || CNpCNqNp
11, 9       13 || CNqp
12, 10      14 || CNqNp
3 p/Nq, q/p 15 || CCNqpCCNqNpNNq
15, 14      16 || CCNqNpNNq
16, 14      17 || NNq
4 p/q       18 || CNNqq
18, 17      19 || q
10-19       20 | CNpq
9-20        21 CpCNpq
21 p/Cpp    22 CCppCNCppq
22, 8       23 CNCppq

Thus, given that $\Gamma$ $\vdash$ NCpp for some formula p, since 23 holds, it follows that $\Gamma$ $\vdash$ q.

Answer 3

With Ax.1 and Ax.2 we can easily prove two useful results : the Deduction Theorem and:

$\vdash (\alpha \to \alpha)$.

From the last one, we have:

$\Gamma, \lnot \beta \vdash (\alpha \to \alpha)$.

From $\Gamma, \vdash \lnot (\alpha \to \alpha)$ we have:

$\Gamma, \lnot \beta \vdash \lnot (\alpha \to \alpha)$.

Now we may "cook" the two results together with the Ded Th and Ax.3:

$\vdash (\delta \to \gamma) \to ((\delta \to \lnot \gamma) \to \lnot \delta)$

to conclude with:

$\Gamma \vdash \lnot \lnot \beta$.

Using Ax.10 (Double Negation):

$\vdash \lnot \lnot \beta \to \beta$,

we conclude with:

$\Gamma \vdash \beta$.