  While non-contextual hidden-variable theories are proved to be impossible,
contextual ones are possible. In a contextual hidden-variable theory, an
observable is called a beable if the hidden-variable assigns its value in a
given measurement context specified by a state and a preferred observable.
Halvorson and Clifton characterized the algebraic structure of beables as a von
Neumann subalgebra, called a beable subalgebra, of the full observable algebra
such that the probability distribution of every observable affiliated therewith
admits the ignorance interpretation. On the other hand, we have shown that for
every von Neumann algebra there is a unique set theoretical universe such that
the internal "real numbers" bijectively correspond to the observables
affiliated with the given von Neumann algebra. Here, we show that a set
theoretical universe is associated with a beable subalgebra if and only if it
is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with
probability equal to unity. Moreover, we show that there is a unique maximal
ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament
and others, by the given measurement context. The set theoretical language for
the ZFC-satisfiable universe, characterized by the present work, rigorously
reconstructs Bohr's notion of the "classical language" to describe the beables
in a given measurement context.
