Full Idea
One of the strengths of ZFC is that it shows that the concept of set is a mathematical concept. Many originally took it to be a logical concept. But ZFC makes mind-boggling existence claims, which should not follow if it was a logical concept.
Gist of Idea
ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
Source
Leon Horsten (The Tarskian Turn [2011], 05.2.3)
Book Reference
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.65
A Reaction
This suggests that set theory is not just a way of expressing mathematics (see Benacerraf 1965), but that some aspect of mathematics has been revealed by it - maybe even its essential nature.