How does maths consistency impacts on computer science?

Question

Lot of mathematicians have made several efforts to answer the question : are mathematics consistent ? Although we haven't yet had a proof of consistency, and surely we will never (Gödel second theorem), does this question has an impact on computer science ? I mean, does the work of those mathematicians, like Gödel's theorems or formal deduction systems, shows new answers or new problems in computer science ? What would happen in computer science if, for example, we found a paradox in set theory ?