Why does the critical value varies inversely with the level of significance in hypothesis testing? The level of significance $\alpha$ is supposed to the probability at which the null hypothesis $H_0$ can be rejected. But when checking the statistical tables for $t_\alpha$, $z_\alpha$, and $\chi^2_\alpha$, the values increase with decrease in $\alpha$ leading to the conception that the acceptance region increases as the strictness increases. 
Eg., in goodness of fit test
$\chi^2_{0.05} = 21.026 \qquad\chi^2_{0.01} = 26.217 \qquad \text{where } \nu=12 $ 
$\alpha=0.01$ means lesser probability of error than $\alpha=0.05$, hence more stricter. But gives a greater range for the acceptance region in the test where $H_0$ is rejected when $\chi^2 > \chi^2_\alpha$.
But this cannot be so. How could one explain this?
 A: 
The level of significance $α$ is supposed to the probability at which the null hypothesis $H_0$ can be rejected. 

Almost. Let's be careful here. Keeping in mind that a type I error is the case where you reject when $H_0$ is true, $\alpha$ is the probability of a type I error under the rejection rule "reject when the statistic is at least as large as the critical value".

But when checking the statistical tables for $t_α$, $z_α$, and $χ^2_α$, the values increase with decrease in $α$

... exactly as they should.

leading to the conception that the acceptance region increases as the strictness increases.

Correct. To have smaller type I error rate (smaller probability of rejection when H0 is true), the rejection region must get smaller ... so the part that isn't the rejection region must get bigger.

$α=0.01$ means lesser probability of error than $α=0.05$

Less probability of a Type I error. The distinction matters here.

But gives a greater range for the acceptance region in the test

Correct -- less chance of a rejection in the case when you shouldn't reject.
