Full Idea
If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory.
Gist of Idea
You can't prove consistency using a weaker theory, but you can use a consistent theory
Source
Øystein Linnebo (Philosophy of Mathematics [2017], 4.6)
Book Reference
Linnebo,Øystein: 'Philosophy of Mathematics' [Princeton 2017], p.71
A Reaction
Note that this concerns consistency, not completeness.