This analysis shows that ineffability in Hλ+-reflecting cardinals implies κ is Hκ+-reflecting, highlighting key properties in set theory.
Suppose κ is a regular cardinal. We prove that if the set of Hλ+-reflecting cardinals λ < κ is ineffable, then κ is an Hκ+-reflecting cardinal. Similarly, we also prove that if the set of Woodin cardinals/cardinals having the stationary reflection property below κ is ineffable, then κ is a Woodin cardinal/cardinal having the stationary reflection property.
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Franqui Cárdenas (2025) studied this question.
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