[FOM] Proposed axiom schemes

joeshipman at aol.com joeshipman at aol.com
Mon Dec 16 12:45:18 EST 2019


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
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