"An essential incorporation of impredicative mathematics in basic physics would involve a revolutionary shift in our understanding of physical reality of a magnitude which would dwarf the passage from classical to quantum mechanics [...} the likelihood of ZFC turning out to be inconsistent [is] much higher than the likelihood of it turning out to be essential to basic physics. The assumption that set-theoretically substantial mathematics is of any use in current science is simply false"
By "impredicative mathematics", he means mathematics with the powerset axiom.
I actually think Weaver misses the essential point, which is this:
The notion of falsifiability, which is the cornerstone of science, can be formalized in such a way that it can be made the cornerstone of mathematics, and it is eminently reasonable to do so; if we don't accept falsifiability as part of the underlying logic of our mathematics, then our mathematics is deficient as a language for science. Impredicative mathematics is not compatible with falsifiability.
The conclusion is that the claim that it is even remotely possible that impredicative mathematics (e.g. ZFC) has an essential role to play in science is a truly extraordinary claim that requires truly extraordinary evidence, and such evidence is woefully lacking.