Corollary 28. ZFC proves CH⇒Θ . For each fixed Φ as above, it also proves CH⇒Φ , and (ZFC) (ZFC+Θ) (ZFC+Φ). If ZFC is consistent, no such Φ is a theorem of ZFC.

[Proof of Corollary 28] The class and strong relation are given by fixed formulas with the set of coding maps as parameter. Expanding these formulas in the AEC axioms and the stated model assertions gives the sentence Θ . The construction and its verification in Theorem 1.1 therefore prove, in ZFC, CH⇒Θ . For each Φ specified above, its ZFC-provable fixed-family instance applies to the parameter witnessing Θ . It would give categoricity in h(ℵ_0) , contradicting the two nonisomorphic models required by Θ . Thus ZFC proves Θ⇒Φ . G\"odel's relative-consistency theorem gives (ZFC)⇒ (ZFC+CH) . Since the latter theory proves Θ , its consistency implies (ZFC+Θ) , and the proved implication Θ⇒Φ then gives (ZFC+Φ) . If Φ were a theorem of consistent ZFC, adjoining Φ could not give a consistent theory.