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Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ Vκ, there exists α < κ such that $(V_\alpha,\in,U\cap V_\alpha)$ is an elementary substructure of $(V_\kappa,\in,U)$.

But usually for a structure, $(A,\sigma,I)$ $A$ would refer to domain, $\sigma$ would be signature and $I$ would be interpretation - so in this case, what are three objects in $( \, \, \, )$ in the quote are referring to?

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For models of (fragments of) set theory, in the structure $( V , \in , A )$ the set $A$ is taken to be a new unary relation such that $A(x)$ holds iff $x \in A$. (So these are technically structures over a different language which has in addition to the binary $\in$ an extra unary relation symbol).

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