Those are entirely different. Peano developed a system for talking about arithmetic in a formalized way. This allowed people to talk about arithmetic in new ways, but it didn’t show that previous formulations of arithmetic were wrong. Godel then built on that to show the limits of arithmetic, which still didn’t invalidate that which came before.
The development of complex numbers as an extension of the real numbers didn’t make work with the real numbers invalid.

When a new scientific model is developed, it supercedes the old model. The old model might still have use, but it’s now known to not actually fit reality. Relativity showed that Newtowns model of the cosmos was wrong: it didn’t extend it or generalize it, it showed that it was inadequately describing reality. Close for human scale problems but ultimately wrong.
And we already know relativity is wrong because it doesn’t match experimental results in quantum mechanics.

Science is our understanding of reality. Reality doesn’t change, but our understanding does.
Because math is a fundamentally different from science, if you know something is true then it’s always true given the assumptions.