[FOM] Definability of R^ in a Boolean-valued universe

[Alexander E. Gutman]

Hello, friends.

Let R be P(N) and let B be a ZFC-definable
complete Boolean algebra which is not sigma-distributive
and therefore ZFC |- not[V(B) |= (R^=R)],
i.e., ZFC proves that, provided x=R^,
the equality x=R is not true in V(B).

Is it possible for R^ be definable in V(B)?
More precisely, can there exist a formula f(x) such that
ZFC |- [V(B) |= (A x)(f(x) <=> x=R^)]?

It seems easy to show that R^ is undefinable in V(B x 2).

  Indeed, since V(B x 2) is isomorphic to V(B) x V,
  definability of R^ in V(B x 2) via a formula f(x)
  would imply ZFC |- [V |= (A x)(f(x) <=> x=R)],
  i.e., ZFC |- (A x)(f(x) <=> x=R),
  whence ZFC |- [V(B) |= (A x)(f(x) <=> x=R)],
  and therefore ZFC |- [V(B) |= (R^=R)].

But what about (un)definability of R^ in V(B)?

-- 
Alexander Gutman