[Paper Review] Universal Indestructibility

Arthur W. Apter, Joel David Hamkins|ArXiv.org|Aug 3, 1998

Advanced Topology and Set TheoryMathematics4 references21 citations

TL;DR

This paper establishes a model of set-theoretic universes in which every supercompact and partially supercompact cardinal is fully indestructible under <κ-directed closed forcing, assuming the existence of a high-jump cardinal. The construction uses a modified forcing iteration that avoids the gaps inherent in Laver's preparation, thereby overcoming the limitations of prior methods and achieving universal indestructibility where previously impossible.

ABSTRACT

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.

Motivation & Objective

• To overcome the limitation of Laver's preparation, which fails to ensure indestructibility for partially supercompact cardinals.
• To construct a model in which all supercompact and partially supercompact cardinals are fully indestructible under <κ-directed closed forcing.
• To show that universal indestructibility is consistent relative to a high-jump cardinal, a large cardinal notion with consistency strength between supercompact and almost huge.
• To demonstrate that the standard Laver preparation cannot achieve universal indestructibility due to its gap structure, which limits the preservation of non-supercompact supercompactness.
• To provide a new forcing iteration strategy that avoids gaps below measurable cardinals, ensuring indestructibility is preserved through later stages.

Proposed method

• A reverse Easton iteration is used, where at each stage γ, a forcing is applied based on a Laver function that selects a <γ-directed closed poset only if it is definable in V_γ.
• The iteration is designed so that no stage of forcing admits a gap below any measurable cardinal γ, preventing the application of the Gap Forcing Theorem.
• The construction ensures that after forcing, any measurable cardinal γ remains indestructible if and only if it was supercompact in the ground model, thus preserving indestructibility for all supercompact cardinals.
• A trial-by-fire argument is used to show that if a cardinal γ is not supercompact in the ground model, then its supercompactness cannot be preserved through the iteration.
• The proof uses a lifting argument with an elementary embedding j: V → M, where M is closed under <θ sequences, to show that if ℚ destroys λ-supercompactness, then a contradiction arises in the lifted model.
• A master condition is constructed below j(g) to lift the embedding through the forcing extension, showing that the measure cannot be added at later stages, thus contradicting the assumption that ℚ destroys supercompactness.

Experimental results

Research questions

• RQ1 Can universal indestructibility for supercompact cardinals be achieved in a model where even partially supercompact cardinals are fully indestructible under <κ-directed closed forcing?
• RQ2 Is it possible to construct such a model without relying on the Laver preparation, which fails to preserve indestructibility for non-supercompact partially supercompact cardinals?
• RQ3 What large cardinal assumption is sufficient to ensure universal indestructibility for all supercompact and partially supercompact cardinals?
• RQ4 Can the Gap Forcing Theorem be circumvented in a forcing iteration to preserve indestructibility for non-supercompact cardinals?
• RQ5 Is universal indestructibility consistent with a single supercompact cardinal, or does it require multiple supercompact cardinals or stronger hypotheses?

Key findings

• Universal indestructibility—where every supercompact and partially supercompact cardinal is fully indestructible under <κ-directed closed forcing—can be achieved in a model with a single supercompact cardinal.
• The existence of a high-jump cardinal is sufficient to construct such a model, establishing the consistency of universal indestructibility relative to this large cardinal hypothesis.
• The Laver preparation fails to achieve universal indestructibility because it admits gaps below measurable cardinals, which by the Gap Forcing Theorem prevent the increase of supercompactness degree.
• Any forcing that admits a gap below γ cannot increase the degree of supercompactness, strong compactness, or strongness of γ, which limits the possibility of universal indestructibility in such models.
• The paper constructs a new forcing iteration that avoids gaps below measurable cardinals, ensuring that all supercompactness properties are preserved through the entire iteration process.
• The key contradiction arises when assuming a forcing destroys λ-supercompactness: the lifted embedding shows the measure must have been present earlier, contradicting the assumption that it was destroyed.