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2 results

  • WEAK INDESTRUCTIBILITY AND REFLECTION

  • Part of
  • JAMES HOLLAND
  • Journal:
    The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
    Published online by Cambridge University Press:
    04 October 2023, pp. 980-1006
    Print publication:
    September 2024
    • Article
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  • We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of the ground model). One direction is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed. This work is a part of my upcoming thesis [7].