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Full Idea
There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure.
Gist of Idea
There is no single agreed structure for set theory
Source
David Bostock (Philosophy of Mathematics [2009], 6.4)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.188
A Reaction
This contrasts with Peano Arithmetic, which is categorical in its second-order version.