[FOM] query regarding omega-completeness of ZFC for Th(L(R))

[Paul Larson]

In response to the posting of Rupert McCallum quoted below:

One way to see this is to consider the set R^#, which codes the theory of L(R) and is universally Baire in this context.
Constructing L(R)^V_kappa_1 and L(R)^V_kappa_2 using R^#, i.e., using kappa_1 or kappa_2 many indiscernibles, one gets elementarily equivalent structures.

A related point is discussed near the bottom of page 16 of the following expository paper: http://www.users.muohio.edu/larsonpb/omega_logic.pdf


>In "On the Question of Absolute Undecidability" Peter Koellner says that assuming a proper class of Woodin cardinals, the theory of L(R) is invariant under>forcing, and then goes on to say that this can be paraphrased as: assuming a proper class of Woodin cardinals ZFC is Omega-complete for the theory of L(R).>This is not obvious to me because one needs to prove that if V_kappa_1 and V_kappa_2 are both models of ZFC, then L(R)^V_kappa_1 and L(R)^V_kappa_2 are>elementarily equivalent. Can anyone explain how this is proved?