By Godel's incompleteness theorems, a formula expressing consistency of a theory that can contain Peano arithmetic cannot be derived or contained from/in the theory.
Godel's completeness theorem states that if a formula $\phi$ is true in every model, then finite deduction for truth of $\phi$ can be made.
With that said, for some reasons, especially by the method of infinitely many deduction steps, we realize that the theory $T$ has to be consistent, which means that formula $\phi_1$ that expresses consistency of the theory $T$ is true. This would mean that every model will satisfy $\phi_1$. However, by Godel's completeness theorem, this would mean that there exists the method of finite deduction that shows $\phi_1$ is true, which would mean that Godel's incompleteness theorem cannot be maintained.
Of course this is obvious, as containing $\phi_1$ would lead to inconsistency as predicted by Godel's incompleteness theorem. My question is, then when theory $T$ can never be shown to be consistent regardless of whether deduction is finite or not, how can we be so sure to believe that $T$ is consistent? Is it because asking whether $\neg \phi_1$ is true is meaningless because if so, theory $T$ itself is completely messed up? (only some deduction/model example would prove that $T$ is inconsistent) Isn't there possibility that the whole argument above shows that $T$ is indeed inconsistent?
By the way, of course infinite steps of deduction is not currently possible. I am rather assuming "as if that is possible, what would happen?"
Finite deduction steps would refer to ordinary finite algorithm.