[FOM] Proposed axiom schemes

[joeshipman at aol.com]

Add a constant Kappa to the language of set theory, and the ZFC axioms, and an axiom stating that Kappa is an ordinal and V_Kappa satisfies ZFC, and for each sentence Phi not referencing Kappa, an axiom saying Phi<-->(Phi holds in V_Kappa).
Kappa could be called an "infallible cardinal"

What is the weakest large cardinal axiom which proves that this formal system is consistent? 
If you have c+ inaccessibles, where c is the cardinality of the continuum, then two inaccessible ranks V_k1 and V_k2 must have the same first order theory where k1<k2, so V_k2 satisfies this axiom scheme when interpreting Kappa as k1. But can you do it with less than c+ inacccessibles?
What happens if you also allow Phi to refer to Kappa?
-- JS
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20191216/f0893d0b/attachment-0001.html>