Doesn't the Gödel theorem say that such proof cannot be found for all conceivable systems; but that for any particular system, a proof could in principle be found?
Things like first order logic are too simple to introduce the kinds of statements Godel used to prove the incompleteness theorem about arithmetic.
And there are proof systems that use this approach (i.e. build a series of provers that each prove the next (more complex) one).
See, for instance, "Coq in Coq", by Bruno Barras and Benjamin Werner, http://www.lix.polytechnique.fr/~barras/publi/coqincoq.pdf , in which the Calculus of Inductive Constructions is used to prove the consistency of the Calculus of Constructions.
See also John Harrison's "Towards self-verification of HOL Light", http://www.cl.cam.ac.uk/~jrh13/papers/holhol.pdf , and Gentzen's consistency proof, https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof , which use similar strategies (by assuming the existence of an inaccessible and assuming recursion up to epsilon_0).
Sometimes one can get relative consistency results. For instance, I believe it is provable in ZF that if ZF is consistent, so if ZF+AC+(V=L)+GCH.