[FOM] Definability of R^ in a Boolean-valued universe
Alexander E. Gutman
gutman at math.nsc.ru
Sat May 4 07:12:19 EDT 2013
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
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